id	sid	tid	token	lemma	pos
cana-843	1	1	communications	communication	NOUN
cana-843	1	2	on	on	ADP
cana-843	1	3	applied	apply	VERB
cana-843	1	4	nonlinear	nonlinear	ADJ
cana-843	1	5	analysis	analysis	NOUN
cana-843	1	6	issn	issn	NOUN
cana-843	1	7	:	:	PUNCT
cana-843	1	8	1074	1074	NUM
cana-843	1	9	-	-	PUNCT
cana-843	1	10	133x	133x	NUM
cana-843	1	11	vol	vol	NOUN
cana-843	1	12	31	31	NUM
cana-843	1	13	no	no	NOUN
cana-843	1	14	.	.	PUNCT
cana-843	2	1	4s	4s	NUM
cana-843	2	2	(	(	PUNCT
cana-843	2	3	2024	2024	NUM
cana-843	2	4	)	)	PUNCT
cana-843	2	5	219	219	NUM
cana-843	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	2	7	the	the	DET
cana-843	2	8	forcing	force	VERB
cana-843	2	9	circular	circular	ADJ
cana-843	2	10	number	number	NOUN
cana-843	2	11	of	of	ADP
cana-843	2	12	a	a	DET
cana-843	2	13	graph	graph	NOUN
cana-843	2	14	1s	1	NOUN
cana-843	2	15	.	.	PUNCT
cana-843	3	1	sheeja	sheeja	ADJ
cana-843	3	2	,	,	PUNCT
cana-843	3	3	2,*k	2,*k	NUM
cana-843	3	4	.	.	PUNCT
cana-843	4	1	rajendran	rajendran	PROPN
cana-843	4	2	1research	1research	NUM
cana-843	4	3	scholar	scholar	NOUN
cana-843	4	4	,	,	PUNCT
cana-843	4	5	vels	vels	PROPN
cana-843	4	6	institute	institute	PROPN
cana-843	4	7	of	of	ADP
cana-843	4	8	science	science	PROPN
cana-843	4	9	technology	technology	NOUN
cana-843	4	10	and	and	CCONJ
cana-843	4	11	advanced	advanced	ADJ
cana-843	4	12	studies	study	NOUN
cana-843	4	13	,	,	PUNCT
cana-843	4	14	chennai	chennai	PROPN
cana-843	4	15	,	,	PUNCT
cana-843	4	16	tamil	tamil	PROPN
cana-843	4	17	nadu	nadu	PROPN
cana-843	4	18	,	,	PUNCT
cana-843	4	19	india	india	PROPN
cana-843	4	20	.	.	PUNCT
cana-843	5	1	2associate	2associate	NUM
cana-843	5	2	professor	professor	NOUN
cana-843	5	3	,	,	PUNCT
cana-843	5	4	vels	vels	PROPN
cana-843	5	5	institute	institute	PROPN
cana-843	5	6	of	of	ADP
cana-843	5	7	science	science	PROPN
cana-843	5	8	technology	technology	NOUN
cana-843	5	9	and	and	CCONJ
cana-843	5	10	advanced	advanced	ADJ
cana-843	5	11	studies	study	NOUN
cana-843	5	12	,	,	PUNCT
cana-843	5	13	chennai	chennai	PROPN
cana-843	5	14	,	,	PUNCT
cana-843	5	15	tamil	tamil	PROPN
cana-843	5	16	nadu	nadu	PROPN
cana-843	5	17	,	,	PUNCT
cana-843	5	18	india	india	PROPN
cana-843	5	19	1	1	NUM
cana-843	5	20	email	email	NOUN
cana-843	5	21	i	i	PROPN
cana-843	5	22	d	d	PROPN
cana-843	5	23	:	:	PUNCT
cana-843	5	24	sheeja1304@gmail.com	sheeja1304@gmail.com	X
cana-843	5	25	2	2	NUM
cana-843	5	26	corresponding	correspond	VERB
cana-843	5	27	author	author	NOUN
cana-843	5	28	:	:	PUNCT
cana-843	5	29	gkrajendra59@gmail.com	gkrajendra59@gmail.com	X
cana-843	5	30	article	article	NOUN
cana-843	5	31	history	history	NOUN
cana-843	5	32	:	:	PUNCT
cana-843	5	33	received	receive	VERB
cana-843	5	34	:	:	PUNCT
cana-843	5	35	18	18	NUM
cana-843	5	36	-	-	PUNCT
cana-843	5	37	04	04	NUM
cana-843	5	38	-	-	PUNCT
cana-843	5	39	2024	2024	NUM
cana-843	5	40	revised	revise	VERB
cana-843	5	41	:	:	PUNCT
cana-843	5	42	08	08	NUM
cana-843	5	43	-	-	SYM
cana-843	5	44	06	06	NUM
cana-843	5	45	-	-	PUNCT
cana-843	5	46	2024	2024	NUM
cana-843	5	47	accepted	accept	VERB
cana-843	5	48	:	:	PUNCT
cana-843	5	49	20	20	NUM
cana-843	5	50	-	-	SYM
cana-843	5	51	06	06	NUM
cana-843	5	52	-	-	PUNCT
cana-843	5	53	2024	2024	NUM
cana-843	5	54	abstract	abstract	NOUN
cana-843	5	55	:	:	PUNCT
cana-843	5	56	let	let	VERB
cana-843	5	57	𝑆	𝑆	PROPN
cana-843	5	58	be	be	AUX
cana-843	5	59	a	a	DET
cana-843	5	60	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	5	61	of	of	ADP
cana-843	5	62	graph	graph	NOUN
cana-843	5	63	𝐺	𝐺	PROPN
cana-843	5	64	and	and	CCONJ
cana-843	5	65	let	let	VERB
cana-843	5	66	𝐺	𝐺	PROPN
cana-843	5	67	be	be	AUX
cana-843	5	68	a	a	DET
cana-843	5	69	connected	connected	ADJ
cana-843	5	70	graph	graph	NOUN
cana-843	5	71	.	.	PUNCT
cana-843	6	1	if	if	SCONJ
cana-843	6	2	𝑆	𝑆	PROPN
cana-843	6	3	is	be	AUX
cana-843	6	4	the	the	DET
cana-843	6	5	only	only	ADJ
cana-843	6	6	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	6	7	that	that	PRON
cana-843	6	8	contains	contain	VERB
cana-843	6	9	𝑇	𝑇	PROPN
cana-843	6	10	,	,	PUNCT
cana-843	6	11	then	then	ADV
cana-843	6	12	a	a	DET
cana-843	6	13	subset	subset	NOUN
cana-843	6	14	𝑇	𝑇	PROPN
cana-843	6	15	⊆	⊆	PROPN
cana-843	6	16	𝑆	𝑆	PROPN
cana-843	6	17	is	be	AUX
cana-843	6	18	referred	refer	VERB
cana-843	6	19	to	to	PART
cana-843	6	20	be	be	AUX
cana-843	6	21	a	a	DET
cana-843	6	22	forcing	forcing	NOUN
cana-843	6	23	subset	subset	NOUN
cana-843	6	24	for	for	ADP
cana-843	6	25	𝑆.	𝑆.	PROPN
cana-843	6	26	a	a	DET
cana-843	6	27	minimum	minimum	ADJ
cana-843	6	28	forcing	forcing	NOUN
cana-843	6	29	subset	subset	NOUN
cana-843	6	30	of	of	ADP
cana-843	6	31	𝑆	𝑆	PROPN
cana-843	6	32	is	be	AUX
cana-843	6	33	a	a	DET
cana-843	6	34	forcing	forcing	NOUN
cana-843	6	35	subset	subset	NOUN
cana-843	6	36	for	for	ADP
cana-843	6	37	𝑆	𝑆	PROPN
cana-843	6	38	of	of	ADP
cana-843	6	39	minimum	minimum	ADJ
cana-843	6	40	cardinality	cardinality	NOUN
cana-843	6	41	.	.	PUNCT
cana-843	7	1	the	the	DET
cana-843	7	2	cardinality	cardinality	NOUN
cana-843	7	3	of	of	ADP
cana-843	7	4	a	a	DET
cana-843	7	5	minimum	minimum	ADJ
cana-843	7	6	forcing	forcing	NOUN
cana-843	7	7	subset	subset	NOUN
cana-843	7	8	of	of	ADP
cana-843	7	9	𝑆	𝑆	PROPN
cana-843	7	10	is	be	AUX
cana-843	7	11	the	the	DET
cana-843	7	12	forcing	force	VERB
cana-843	7	13	circular	circular	ADJ
cana-843	7	14	number	number	NOUN
cana-843	7	15	of	of	ADP
cana-843	7	16	𝑆	𝑆	PROPN
cana-843	7	17	,	,	PUNCT
cana-843	7	18	represented	represent	VERB
cana-843	7	19	by	by	ADP
cana-843	7	20	the	the	DET
cana-843	7	21	notation	notation	NOUN
cana-843	7	22	𝑓𝑐𝑟(𝑆	𝑓𝑐𝑟(𝑆	NUM
cana-843	7	23	)	)	PUNCT
cana-843	7	24	.	.	PUNCT
cana-843	8	1	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	8	2	)	)	PUNCT
cana-843	8	3	=	=	SYM
cana-843	8	4	min	min	PROPN
cana-843	8	5	{	{	PUNCT
cana-843	8	6	𝑓𝑐𝑟(𝑆	𝑓𝑐𝑟(𝑆	NUM
cana-843	8	7	)	)	PUNCT
cana-843	8	8	}	}	PUNCT
cana-843	8	9	is	be	AUX
cana-843	8	10	the	the	DET
cana-843	8	11	forcing	force	VERB
cana-843	8	12	circular	circular	ADJ
cana-843	8	13	number	number	NOUN
cana-843	8	14	of	of	ADP
cana-843	8	15	𝐺	𝐺	PROPN
cana-843	8	16	,	,	PUNCT
cana-843	8	17	where	where	SCONJ
cana-843	8	18	the	the	DET
cana-843	8	19	minimum	minimum	NOUN
cana-843	8	20	is	be	AUX
cana-843	8	21	the	the	DET
cana-843	8	22	sum	sum	NOUN
cana-843	8	23	of	of	ADP
cana-843	8	24	all	all	DET
cana-843	8	25	minimum	minimum	ADJ
cana-843	8	26	forcing	force	VERB
cana-843	8	27	circular	circular	ADJ
cana-843	8	28	-	-	PUNCT
cana-843	8	29	sets	set	NOUN
cana-843	8	30	𝑆	𝑆	PROPN
cana-843	8	31	in	in	ADP
cana-843	8	32	𝐺.	𝐺.	NOUN
cana-843	8	33	for	for	ADP
cana-843	8	34	several	several	ADJ
cana-843	8	35	standard	standard	ADJ
cana-843	8	36	graphs	graph	NOUN
cana-843	8	37	,	,	PUNCT
cana-843	8	38	the	the	DET
cana-843	8	39	forcing	force	VERB
cana-843	8	40	circular	circular	ADJ
cana-843	8	41	number	number	NOUN
cana-843	8	42	is	be	AUX
cana-843	8	43	identified	identify	VERB
cana-843	8	44	.	.	PUNCT
cana-843	9	1	it	it	PRON
cana-843	9	2	is	be	AUX
cana-843	9	3	demonstrated	demonstrate	VERB
cana-843	9	4	that	that	SCONJ
cana-843	9	5	there	there	PRON
cana-843	9	6	exists	exist	VERB
cana-843	9	7	a	a	DET
cana-843	9	8	connected	connected	ADJ
cana-843	9	9	graph	graph	NOUN
cana-843	9	10	g	g	ADP
cana-843	9	11	such	such	ADJ
cana-843	9	12	that	that	DET
cana-843	9	13	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	9	14	)	)	PUNCT
cana-843	9	15	=	=	SYM
cana-843	9	16	𝑎	𝑎	PROPN
cana-843	9	17	and	and	CCONJ
cana-843	9	18	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	9	19	)	)	PUNCT
cana-843	10	1	=	=	SYM
cana-843	10	2	𝑏	𝑏	NOUN
cana-843	10	3	for	for	ADP
cana-843	10	4	every	every	DET
cana-843	10	5	integer	integer	NOUN
cana-843	10	6	𝑎	𝑎	PRON
cana-843	10	7	≥	≥	NOUN
cana-843	10	8	0	0	NUM
cana-843	10	9	,	,	PUNCT
cana-843	10	10	and	and	CCONJ
cana-843	10	11	𝑏	𝑏	PRON
cana-843	10	12	≥	≥	NOUN
cana-843	10	13	0	0	NUM
cana-843	10	14	.	.	PUNCT
cana-843	11	1	keywords	keyword	NOUN
cana-843	11	2	:	:	PUNCT
cana-843	11	3	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	11	4	,	,	PUNCT
cana-843	11	5	circular	circular	ADJ
cana-843	11	6	number	number	NOUN
cana-843	11	7	,	,	PUNCT
cana-843	11	8	forcing	force	VERB
cana-843	11	9	circular	circular	ADJ
cana-843	11	10	number	number	NOUN
cana-843	11	11	.	.	PUNCT
cana-843	12	1	ams	am	NOUN
cana-843	12	2	subject	subject	ADJ
cana-843	12	3	classification	classification	NOUN
cana-843	12	4	:	:	PUNCT
cana-843	12	5	05c12	05c12	NOUN
cana-843	12	6	.	.	PUNCT
cana-843	13	1	1	1	NUM
cana-843	13	2	.	.	X
cana-843	13	3	introduction	introduction	NOUN
cana-843	13	4	and	and	CCONJ
cana-843	13	5	preliminaries	preliminary	NOUN
cana-843	13	6	a	a	DET
cana-843	13	7	graph	graph	NOUN
cana-843	13	8	𝐺	𝐺	NOUN
cana-843	13	9	=	=	SYM
cana-843	13	10	(	(	PUNCT
cana-843	13	11	𝑉	𝑉	PROPN
cana-843	13	12	,	,	PUNCT
cana-843	13	13	𝐸	𝐸	PROPN
cana-843	13	14	)	)	PUNCT
cana-843	13	15	is	be	AUX
cana-843	13	16	a	a	DET
cana-843	13	17	connected	connect	VERB
cana-843	13	18	,	,	PUNCT
cana-843	13	19	finite	finite	ADJ
cana-843	13	20	graph	graph	NOUN
cana-843	13	21	that	that	PRON
cana-843	13	22	does	do	AUX
cana-843	13	23	not	not	PART
cana-843	13	24	have	have	VERB
cana-843	13	25	loops	loop	NOUN
cana-843	13	26	or	or	CCONJ
cana-843	13	27	numerous	numerous	ADJ
cana-843	13	28	edges	edge	NOUN
cana-843	13	29	.	.	PUNCT
cana-843	14	1	𝐺	𝐺	PROPN
cana-843	14	2	is	be	AUX
cana-843	14	3	represented	represent	VERB
cana-843	14	4	by	by	ADP
cana-843	14	5	the	the	DET
cana-843	14	6	symbols	symbol	NOUN
cana-843	14	7	𝑛	𝑛	PROPN
cana-843	14	8	and	and	CCONJ
cana-843	14	9	𝑚	𝑚	PROPN
cana-843	14	10	,	,	PUNCT
cana-843	14	11	respectively	respectively	ADV
cana-843	14	12	,	,	PUNCT
cana-843	14	13	for	for	ADP
cana-843	14	14	order	order	NOUN
cana-843	14	15	and	and	CCONJ
cana-843	14	16	size	size	NOUN
cana-843	14	17	.	.	PUNCT
cana-843	15	1	we	we	PRON
cana-843	15	2	use	use	VERB
cana-843	15	3	[	[	X
cana-843	15	4	1,6	1,6	NUM
cana-843	15	5	]	]	PUNCT
cana-843	15	6	for	for	ADP
cana-843	15	7	basic	basic	ADJ
cana-843	15	8	terminology	terminology	NOUN
cana-843	15	9	in	in	ADP
cana-843	15	10	graph	graph	NOUN
cana-843	15	11	theoretic	theoretic	NOUN
cana-843	15	12	.	.	PUNCT
cana-843	16	1	if	if	SCONJ
cana-843	16	2	𝑢𝑣	𝑢𝑣	PROPN
cana-843	16	3	∈	∈	PROPN
cana-843	16	4	𝐸(𝐺	𝐸(𝐺	NOUN
cana-843	16	5	)	)	PUNCT
cana-843	16	6	,	,	PUNCT
cana-843	16	7	“	"	PUNCT
cana-843	16	8	then	then	ADV
cana-843	16	9	two	two	NUM
cana-843	16	10	vertices	vertex	NOUN
cana-843	16	11	,	,	PUNCT
cana-843	16	12	𝑢	𝑢	NOUN
cana-843	16	13	and	and	CCONJ
cana-843	16	14	𝑣	𝑣	ADP
cana-843	16	15	,	,	PUNCT
cana-843	16	16	are	be	AUX
cana-843	16	17	considered	consider	VERB
cana-843	16	18	nearby	nearby	ADV
cana-843	16	19	in	in	ADP
cana-843	16	20	𝐺.	𝐺.	NOUN
cana-843	16	21	the	the	DET
cana-843	16	22	collection	collection	NOUN
cana-843	16	23	of	of	ADP
cana-843	16	24	vertices	vertex	NOUN
cana-843	16	25	next	next	ADV
cana-843	16	26	to	to	ADP
cana-843	16	27	a	a	DET
cana-843	16	28	vertex	vertex	NOUN
cana-843	16	29	𝑣	𝑣	ADP
cana-843	16	30	in	in	ADP
cana-843	16	31	𝐺	𝐺	PROPN
cana-843	16	32	is	be	AUX
cana-843	16	33	called	call	VERB
cana-843	16	34	its	its	PRON
cana-843	16	35	neighbourhood	neighbourhood	NOUN
cana-843	16	36	,	,	PUNCT
cana-843	16	37	or	or	CCONJ
cana-843	16	38	𝑁(𝑣	𝑁(𝑣	NUM
cana-843	16	39	)	)	PUNCT
cana-843	16	40	.	.	PUNCT
cana-843	17	1	the	the	DET
cana-843	17	2	vertex	vertex	NOUN
cana-843	17	3	𝑣	𝑣	PART
cana-843	17	4	has	have	VERB
cana-843	17	5	a	a	DET
cana-843	17	6	degree	degree	NOUN
cana-843	17	7	of	of	ADP
cana-843	17	8	𝑑𝑒𝑔(𝑣	𝑑𝑒𝑔(𝑣	NOUN
cana-843	17	9	)	)	PUNCT
cana-843	17	10	=	=	SYM
cana-843	17	11	|𝑁(𝑣)|	|𝑁(𝑣)|	PROPN
cana-843	17	12	.	.	PUNCT
cana-843	18	1	we	we	PRON
cana-843	18	2	refer	refer	VERB
cana-843	18	3	to	to	ADP
cana-843	18	4	u	u	NOUN
cana-843	18	5	as	as	ADP
cana-843	18	6	an	an	DET
cana-843	18	7	end	end	NOUN
cana-843	18	8	edge	edge	NOUN
cana-843	18	9	,	,	PUNCT
cana-843	18	10	u	u	NOUN
cana-843	18	11	as	as	ADP
cana-843	18	12	a	a	DET
cana-843	18	13	leaf	leaf	NOUN
cana-843	18	14	,	,	PUNCT
cana-843	18	15	and	and	CCONJ
cana-843	18	16	v	v	NOUN
cana-843	18	17	as	as	ADP
cana-843	18	18	a	a	DET
cana-843	18	19	support	support	NOUN
cana-843	18	20	vertex	vertex	NOUN
cana-843	18	21	if	if	SCONJ
cana-843	18	22	𝑒	𝑒	PROPN
cana-843	18	23	=	=	PUNCT
cana-843	18	24	{	{	PUNCT
cana-843	18	25	𝑢	𝑢	X
cana-843	18	26	,	,	PUNCT
cana-843	18	27	𝑣	𝑣	PRON
cana-843	18	28	}	}	PUNCT
cana-843	18	29	is	be	AUX
cana-843	18	30	an	an	DET
cana-843	18	31	edge	edge	NOUN
cana-843	18	32	of	of	ADP
cana-843	18	33	a	a	DET
cana-843	18	34	graph	graph	NOUN
cana-843	18	35	g	g	NOUN
cana-843	18	36	with	with	ADP
cana-843	18	37	𝑑𝑒𝑔(𝑢	𝑑𝑒𝑔(𝑢	PROPN
cana-843	18	38	)	)	PUNCT
cana-843	18	39	=	=	SYM
cana-843	18	40	1	1	NUM
cana-843	18	41	and	and	CCONJ
cana-843	18	42	𝑑𝑒𝑔(𝑣	𝑑𝑒𝑔(𝑣	NOUN
cana-843	18	43	)	)	PUNCT
cana-843	18	44	>	>	X
cana-843	19	1	1	1	X
cana-843	19	2	.	.	PUNCT
cana-843	19	3	the	the	DET
cana-843	19	4	greatest	great	ADJ
cana-843	19	5	degree	degree	NOUN
cana-843	19	6	of	of	ADP
cana-843	19	7	a	a	DET
cana-843	19	8	graph	graph	NOUN
cana-843	19	9	𝐺	𝐺	NOUN
cana-843	19	10	is	be	AUX
cana-843	19	11	shown	show	VERB
cana-843	19	12	by	by	ADP
cana-843	19	13	∆(𝐺	∆(𝐺	PROPN
cana-843	19	14	)	)	PUNCT
cana-843	19	15	.	.	PUNCT
cana-843	20	1	𝐺[𝑆	𝐺[𝑆	NOUN
cana-843	20	2	]	]	PUNCT
cana-843	20	3	is	be	AUX
cana-843	20	4	the	the	DET
cana-843	20	5	representation	representation	NOUN
cana-843	20	6	of	of	ADP
cana-843	20	7	the	the	DET
cana-843	20	8	subgraph	subgraph	NOUN
cana-843	20	9	that	that	PRON
cana-843	20	10	a	a	DET
cana-843	20	11	set	set	ADJ
cana-843	20	12	𝑆	𝑆	PROPN
cana-843	20	13	of	of	ADP
cana-843	20	14	vertices	vertex	NOUN
cana-843	20	15	of	of	ADP
cana-843	20	16	a	a	DET
cana-843	20	17	graph	graph	NOUN
cana-843	20	18	𝐺	𝐺	NOUN
cana-843	20	19	induces	induce	NOUN
cana-843	20	20	,	,	PUNCT
cana-843	20	21	where	where	SCONJ
cana-843	20	22	𝑉	𝑉	PROPN
cana-843	20	23	(	(	PUNCT
cana-843	20	24	𝐺[𝑆	𝐺[𝑆	NOUN
cana-843	20	25	]	]	PUNCT
cana-843	20	26	)	)	PUNCT
cana-843	20	27	=	=	SYM
cana-843	20	28	𝑆	𝑆	PROPN
cana-843	20	29	and	and	CCONJ
cana-843	20	30	𝐸(𝐺[𝑆	𝐸(𝐺[𝑆	ADJ
cana-843	20	31	]	]	PUNCT
cana-843	20	32	)	)	PUNCT
cana-843	20	33	=	=	PRON
cana-843	20	34	{	{	PUNCT
cana-843	20	35	𝑢𝑣	𝑢𝑣	NOUN
cana-843	20	36	∈	∈	PROPN
cana-843	20	37	𝐸(𝐺	𝐸(𝐺	NOUN
cana-843	20	38	)	)	PUNCT
cana-843	20	39	∶	∶	NOUN
cana-843	20	40	𝑢	𝑢	X
cana-843	20	41	,	,	PUNCT
cana-843	20	42	𝑣	𝑣	PROPN
cana-843	20	43	∈	∈	PROPN
cana-843	20	44	𝑆	𝑆	PROPN
cana-843	20	45	}	}	PUNCT
cana-843	20	46	.	.	PUNCT
cana-843	21	1	a	a	DET
cana-843	21	2	vertex	vertex	NOUN
cana-843	21	3	𝑣	𝑣	NOUN
cana-843	21	4	is	be	AUX
cana-843	21	5	an	an	DET
cana-843	21	6	extreme	extreme	ADJ
cana-843	21	7	vertex	vertex	NOUN
cana-843	21	8	of	of	ADP
cana-843	21	9	𝐺	𝐺	PROPN
cana-843	21	10	if	if	SCONJ
cana-843	21	11	and	and	CCONJ
cana-843	21	12	only	only	ADV
cana-843	21	13	if	if	SCONJ
cana-843	21	14	𝐺[𝑁(𝑣	𝐺[𝑁(𝑣	VERB
cana-843	21	15	)	)	PUNCT
cana-843	21	16	]	]	PUNCT
cana-843	21	17	is	be	AUX
cana-843	21	18	complete	complete	ADJ
cana-843	21	19	.	.	PUNCT
cana-843	22	1	the	the	DET
cana-843	22	2	length	length	NOUN
cana-843	22	3	of	of	ADP
cana-843	22	4	the	the	DET
cana-843	22	5	shortest	short	ADJ
cana-843	22	6	path	path	NOUN
cana-843	22	7	between	between	ADP
cana-843	22	8	two	two	NUM
cana-843	22	9	vertices	vertex	NOUN
cana-843	22	10	𝑢	𝑢	PART
cana-843	22	11	,	,	PUNCT
cana-843	22	12	𝑣	𝑣	PRON
cana-843	22	13	∈	∈	PROPN
cana-843	22	14	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	22	15	)	)	PUNCT
cana-843	22	16	is	be	AUX
cana-843	22	17	the	the	DET
cana-843	22	18	distance	distance	NOUN
cana-843	22	19	𝑑(𝑢	𝑑(𝑢	NOUN
cana-843	22	20	,	,	PUNCT
cana-843	22	21	𝑣	𝑣	NOUN
cana-843	22	22	)	)	PUNCT
cana-843	22	23	.	.	PUNCT
cana-843	23	1	a	a	DET
cana-843	23	2	𝑢	𝑢	NOUN
cana-843	23	3	−	−	NOUN
cana-843	23	4	𝑣	𝑣	DET
cana-843	23	5	geodesic	geodesic	NOUN
cana-843	23	6	of	of	ADP
cana-843	23	7	𝐺	𝐺	PROPN
cana-843	23	8	is	be	AUX
cana-843	23	9	any	any	DET
cana-843	23	10	𝑢	𝑢	NOUN
cana-843	23	11	−	−	NOUN
cana-843	23	12	𝑣	𝑣	PRON
cana-843	23	13	path	path	NOUN
cana-843	23	14	of	of	ADP
cana-843	23	15	length	length	NOUN
cana-843	23	16	𝑑(𝑢	𝑑(𝑢	NOUN
cana-843	23	17	,	,	PUNCT
cana-843	23	18	𝑣	𝑣	NOUN
cana-843	23	19	)	)	PUNCT
cana-843	23	20	.	.	PUNCT
cana-843	24	1	if	if	SCONJ
cana-843	24	2	𝑥	𝑥	PROPN
cana-843	24	3	is	be	AUX
cana-843	24	4	a	a	DET
cana-843	24	5	vertex	vertex	NOUN
cana-843	24	6	of	of	ADP
cana-843	24	7	𝑃	𝑃	NOUN
cana-843	24	8	and	and	CCONJ
cana-843	24	9	𝑥	𝑥	NOUN
cana-843	24	10	≠	≠	PROPN
cana-843	24	11	𝑢	𝑢	PROPN
cana-843	24	12	,	,	PUNCT
cana-843	24	13	𝑣	𝑣	NOUN
cana-843	24	14	,	,	PUNCT
cana-843	24	15	then	then	ADV
cana-843	24	16	𝑥	𝑥	PROPN
cana-843	24	17	is	be	AUX
cana-843	24	18	an	an	DET
cana-843	24	19	internal	internal	ADJ
cana-843	24	20	vertex	vertex	NOUN
cana-843	24	21	of	of	ADP
cana-843	24	22	a	a	DET
cana-843	24	23	e𝑢−	e𝑢−	PROPN
cana-843	24	24	e𝑣	e𝑣	PROPN
cana-843	24	25	path	path	PROPN
cana-843	24	26	𝑃.	𝑃.	PROPN
cana-843	24	27	𝐼[e𝑢	𝐼[e𝑢	PROPN
cana-843	24	28	,	,	PUNCT
cana-843	24	29	e𝑣	e𝑣	PROPN
cana-843	24	30	]	]	PUNCT
cana-843	24	31	is	be	AUX
cana-843	24	32	the	the	DET
cana-843	24	33	closed	closed	ADJ
cana-843	24	34	interval	interval	NOUN
cana-843	24	35	consisting	consist	VERB
cana-843	24	36	of	of	ADP
cana-843	24	37	𝑢	𝑢	NOUN
cana-843	24	38	,	,	PUNCT
cana-843	24	39	𝑣	𝑣	X
cana-843	24	40	and	and	CCONJ
cana-843	24	41	all	all	DET
cana-843	24	42	vertices	vertex	NOUN
cana-843	24	43	that	that	PRON
cana-843	24	44	are	be	AUX
cana-843	24	45	on	on	ADP
cana-843	24	46	a	a	DET
cana-843	24	47	e𝑢−	e𝑢−	PROPN
cana-843	24	48	e𝑣	e𝑣	PROPN
cana-843	24	49	geodesic	geodesic	NOUN
cana-843	24	50	of	of	ADP
cana-843	24	51	𝐺.	𝐺.	NOUN
cana-843	24	52	the	the	DET
cana-843	24	53	closure	closure	NOUN
cana-843	24	54	of	of	ADP
cana-843	24	55	a	a	DET
cana-843	24	56	non	non	ADJ
cana-843	24	57	-	-	ADJ
cana-843	24	58	empty	empty	ADJ
cana-843	24	59	set	set	VERB
cana-843	24	60	𝑆	𝑆	PROPN
cana-843	24	61	⊆	⊆	NUM
cana-843	24	62	𝑉	𝑉	PROPN
cana-843	24	63	(	(	PUNCT
cana-843	24	64	𝐺	𝐺	NOUN
cana-843	24	65	)	)	PUNCT
cana-843	24	66	is	be	AUX
cana-843	24	67	given	give	VERB
cana-843	24	68	by	by	ADP
cana-843	24	69	the	the	DET
cana-843	24	70	set	set	NOUN
cana-843	24	71	𝐼[𝑆	𝐼[𝑆	NOUN
cana-843	24	72	]	]	X
cana-843	24	73	=	=	SYM
cana-843	24	74	⋃	⋃	NOUN
cana-843	24	75	𝐼[𝑢	𝐼[𝑢	X
cana-843	24	76	,	,	PUNCT
cana-843	24	77	𝑣]𝑢,𝑣∈𝑆	𝑣]𝑢,𝑣∈𝑆	PROPN
cana-843	24	78	.	.	PUNCT
cana-843	25	1	if	if	SCONJ
cana-843	25	2	𝐼[𝑆	𝐼[𝑆	NOUN
cana-843	25	3	]	]	X
cana-843	25	4	=	=	SYM
cana-843	25	5	𝑉	𝑉	PROPN
cana-843	25	6	(	(	PUNCT
cana-843	25	7	𝐺	𝐺	NOUN
cana-843	25	8	)	)	PUNCT
cana-843	25	9	,	,	PUNCT
cana-843	25	10	then	then	ADV
cana-843	25	11	a	a	DET
cana-843	25	12	set	set	NOUN
cana-843	25	13	𝑆	𝑆	PROPN
cana-843	25	14	⊆	⊆	NUM
cana-843	25	15	𝑉e(𝐺	𝑉e(𝐺	NUM
cana-843	25	16	)	)	PUNCT
cana-843	25	17	is	be	AUX
cana-843	25	18	a	a	DET
cana-843	25	19	geodetic	geodetic	ADJ
cana-843	25	20	set	set	NOUN
cana-843	25	21	.	.	PUNCT
cana-843	26	1	the	the	DET
cana-843	26	2	geodetic	geodetic	ADJ
cana-843	26	3	number	number	NOUN
cana-843	26	4	of	of	ADP
cana-843	26	5	𝐺	𝐺	PROPN
cana-843	26	6	,	,	PUNCT
cana-843	26	7	represented	represent	VERB
cana-843	26	8	by	by	ADP
cana-843	26	9	𝑔(𝐺	𝑔(𝐺	PROPN
cana-843	26	10	)	)	PUNCT
cana-843	26	11	,	,	PUNCT
cana-843	26	12	is	be	AUX
cana-843	26	13	the	the	DET
cana-843	26	14	lowest	low	ADJ
cana-843	26	15	cardinality	cardinality	NOUN
cana-843	26	16	of	of	ADP
cana-843	26	17	a	a	DET
cana-843	26	18	geodetic	geodetic	ADJ
cana-843	26	19	set	set	NOUN
cana-843	26	20	of	of	ADP
cana-843	26	21	𝐺.	𝐺.	NOUN
cana-843	26	22	a	a	DET
cana-843	26	23	𝑔	𝑔	NOUN
cana-843	26	24	−set	−set	NOUN
cana-843	26	25	of	of	ADP
cana-843	26	26	𝐺	𝐺	PROPN
cana-843	26	27	is	be	AUX
cana-843	26	28	a	a	DET
cana-843	26	29	geodetic	geodetic	ADJ
cana-843	26	30	set	set	NOUN
cana-843	26	31	”	"	PUNCT
cana-843	26	32	of	of	ADP
cana-843	26	33	minimum	minimum	ADJ
cana-843	26	34	cardinality	cardinality	NOUN
cana-843	26	35	.	.	PUNCT
cana-843	27	1	see	see	VERB
cana-843	28	1	[	[	X
cana-843	28	2	3,4,8	3,4,8	NOUN
cana-843	28	3	]	]	PUNCT
cana-843	28	4	for	for	ADP
cana-843	28	5	references	reference	NOUN
cana-843	28	6	on	on	ADP
cana-843	28	7	geodetic	geodetic	ADJ
cana-843	28	8	parameters	parameter	NOUN
cana-843	28	9	in	in	ADP
cana-843	28	10	graphs	graph	NOUN
cana-843	28	11	.	.	PUNCT
cana-843	29	1	the	the	DET
cana-843	29	2	longest	long	ADJ
cana-843	29	3	path	path	NOUN
cana-843	29	4	between	between	ADP
cana-843	29	5	two	two	NUM
cana-843	29	6	vertices	vertex	NOUN
cana-843	29	7	𝑢	𝑢	PART
cana-843	29	8	,	,	PUNCT
cana-843	29	9	𝑣	𝑣	PRON
cana-843	29	10	∈	∈	PROPN
cana-843	29	11	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	29	12	)	)	PUNCT
cana-843	29	13	is	be	AUX
cana-843	29	14	the	the	DET
cana-843	29	15	detour	detour	NOUN
cana-843	29	16	distance	distance	NOUN
cana-843	29	17	𝐷(𝑢	𝐷(𝑢	NOUN
cana-843	29	18	,	,	PUNCT
cana-843	29	19	𝑣	𝑣	NOUN
cana-843	29	20	)	)	PUNCT
cana-843	29	21	.	.	PUNCT
cana-843	30	1	a	a	DET
cana-843	30	2	𝑢	𝑢	NOUN
cana-843	30	3	−	−	NOUN
cana-843	30	4	𝑣	𝑣	DET
cana-843	30	5	detour	detour	NOUN
cana-843	30	6	of	of	ADP
cana-843	30	7	𝐺	𝐺	PROPN
cana-843	30	8	is	be	AUX
cana-843	30	9	any	any	DET
cana-843	30	10	a𝑢−	a𝑢−	NUM
cana-843	30	11	a𝑣	a𝑣	PRON
cana-843	30	12	path	path	NOUN
cana-843	30	13	of	of	ADP
cana-843	30	14	length	length	PROPN
cana-843	30	15	𝐷(a𝑢	𝐷(a𝑢	PROPN
cana-843	30	16	,	,	PUNCT
cana-843	30	17	a𝑣	a𝑣	PROPN
cana-843	30	18	)	)	PUNCT
cana-843	30	19	.	.	PUNCT
cana-843	31	1	all	all	DET
cana-843	31	2	vertices	vertex	NOUN
cana-843	31	3	of	of	ADP
cana-843	31	4	the	the	DET
cana-843	31	5	closed	closed	ADJ
cana-843	31	6	interval	interval	NOUN
cana-843	31	7	𝐼𝐷[𝑢	𝐼𝐷[𝑢	NOUN
cana-843	31	8	,	,	PUNCT
cana-843	31	9	𝑣	𝑣	NOUN
cana-843	31	10	]	]	X
cana-843	31	11	lie	lie	NOUN
cana-843	31	12	on	on	ADP
cana-843	31	13	some	some	DET
cana-843	31	14	𝑢	𝑢	NOUN
cana-843	31	15	−	−	NOUN
cana-843	31	16	𝑣	𝑣	ADP
cana-843	31	17	communications	communication	NOUN
cana-843	31	18	on	on	ADP
cana-843	31	19	applied	apply	VERB
cana-843	31	20	nonlinear	nonlinear	ADJ
cana-843	31	21	analysis	analysis	NOUN
cana-843	31	22	issn	issn	NOUN
cana-843	31	23	:	:	PUNCT
cana-843	31	24	1074	1074	NUM
cana-843	31	25	-	-	PUNCT
cana-843	31	26	133x	133x	NUM
cana-843	31	27	vol	vol	NOUN
cana-843	31	28	31	31	NUM
cana-843	31	29	no	no	NOUN
cana-843	31	30	.	.	PUNCT
cana-843	32	1	4s	4s	NUM
cana-843	32	2	(	(	PUNCT
cana-843	32	3	2024	2024	NUM
cana-843	32	4	)	)	PUNCT
cana-843	32	5	220	220	NUM
cana-843	32	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	32	7	detour	detour	NOUN
cana-843	32	8	of	of	ADP
cana-843	32	9	𝐺	𝐺	PROPN
cana-843	32	10	,	,	PUNCT
cana-843	32	11	and	and	CCONJ
cana-843	32	12	the	the	DET
cana-843	32	13	interval	interval	NOUN
cana-843	32	14	itself	itself	PRON
cana-843	32	15	consists	consist	VERB
cana-843	32	16	of	of	ADP
cana-843	32	17	𝑢	𝑢	NOUN
cana-843	32	18	,	,	PUNCT
cana-843	32	19	𝑣.	𝑣.	VERB
cana-843	32	20	the	the	DET
cana-843	32	21	closure	closure	NOUN
cana-843	32	22	of	of	ADP
cana-843	32	23	a	a	DET
cana-843	32	24	non	non	ADJ
cana-843	32	25	-	-	ADJ
cana-843	32	26	empty	empty	ADJ
cana-843	32	27	set	set	VERB
cana-843	32	28	𝑆	𝑆	PROPN
cana-843	32	29	⊆	⊆	NUM
cana-843	32	30	𝑉	𝑉	PROPN
cana-843	32	31	(	(	PUNCT
cana-843	32	32	𝐺	𝐺	NOUN
cana-843	32	33	)	)	PUNCT
cana-843	32	34	is	be	AUX
cana-843	32	35	given	give	VERB
cana-843	32	36	by	by	ADP
cana-843	32	37	the	the	DET
cana-843	32	38	set	set	NOUN
cana-843	32	39	𝐼𝐷[𝑆	𝐼𝐷[𝑆	NOUN
cana-843	32	40	]	]	X
cana-843	32	41	=	=	SYM
cana-843	32	42	⋃	⋃	NOUN
cana-843	32	43	𝐼𝐷[𝑢	𝐼𝐷[𝑢	NOUN
cana-843	32	44	,	,	PUNCT
cana-843	32	45	𝑣]𝑢,𝑣∈𝑆	𝑣]𝑢,𝑣∈𝑆	PROPN
cana-843	32	46	.	.	PUNCT
cana-843	33	1	a	a	DET
cana-843	33	2	detour	detour	NOUN
cana-843	33	3	set	set	NOUN
cana-843	33	4	is	be	AUX
cana-843	33	5	then	then	ADV
cana-843	33	6	defined	define	VERB
cana-843	33	7	as	as	ADP
cana-843	33	8	a	a	DET
cana-843	33	9	set	set	NOUN
cana-843	33	10	𝑆	𝑆	PROPN
cana-843	33	11	⊆	⊆	NUM
cana-843	33	12	𝑉	𝑉	PROPN
cana-843	33	13	(	(	PUNCT
cana-843	33	14	𝐺	𝐺	NOUN
cana-843	33	15	)	)	PUNCT
cana-843	33	16	.	.	PUNCT
cana-843	34	1	the	the	DET
cana-843	34	2	detour	detour	NOUN
cana-843	34	3	number	number	NOUN
cana-843	34	4	of	of	ADP
cana-843	34	5	g	g	NOUN
cana-843	34	6	,	,	PUNCT
cana-843	34	7	represented	represent	VERB
cana-843	34	8	by	by	ADP
cana-843	34	9	𝑑𝑛(𝐺	𝑑𝑛(𝐺	NOUN
cana-843	34	10	)	)	PUNCT
cana-843	34	11	,	,	PUNCT
cana-843	34	12	is	be	AUX
cana-843	34	13	the	the	DET
cana-843	34	14	lowest	low	ADJ
cana-843	34	15	cardinality	cardinality	NOUN
cana-843	34	16	of	of	ADP
cana-843	34	17	a	a	DET
cana-843	34	18	detour	detour	NOUN
cana-843	34	19	set	set	NOUN
cana-843	34	20	of	of	ADP
cana-843	34	21	𝐺.	𝐺.	NOUN
cana-843	34	22	a	a	DET
cana-843	34	23	𝑑𝑛-set	𝑑𝑛-set	NOUN
cana-843	34	24	of	of	ADP
cana-843	34	25	𝐺	𝐺	PROPN
cana-843	34	26	is	be	AUX
cana-843	34	27	a	a	DET
cana-843	34	28	diversion	diversion	NOUN
cana-843	34	29	set	set	VERB
cana-843	34	30	with	with	ADP
cana-843	34	31	minimum	minimum	ADJ
cana-843	34	32	cardinality	cardinality	NOUN
cana-843	34	33	.	.	PUNCT
cana-843	35	1	hence	hence	ADV
cana-843	35	2	[	[	X
cana-843	35	3	5,7	5,7	NUM
cana-843	35	4	]	]	PUNCT
cana-843	35	5	covered	cover	VERB
cana-843	35	6	the	the	DET
cana-843	35	7	study	study	NOUN
cana-843	35	8	of	of	ADP
cana-843	35	9	these	these	DET
cana-843	35	10	ideas	idea	NOUN
cana-843	35	11	.	.	PUNCT
cana-843	36	1	𝐷𝑐(𝑢	𝐷𝑐(𝑢	NOUN
cana-843	36	2	,	,	PUNCT
cana-843	36	3	𝑣	𝑣	NOUN
cana-843	36	4	)	)	PUNCT
cana-843	36	5	represents	represent	VERB
cana-843	36	6	the	the	DET
cana-843	36	7	circular	circular	ADJ
cana-843	36	8	distance	distance	NOUN
cana-843	36	9	between	between	ADP
cana-843	36	10	𝑢	𝑢	NOUN
cana-843	36	11	and	and	CCONJ
cana-843	36	12	𝑣	𝑣	ADP
cana-843	36	13	,	,	PUNCT
cana-843	36	14	which	which	PRON
cana-843	36	15	is	be	AUX
cana-843	36	16	represented	represent	VERB
cana-843	36	17	as	as	ADP
cana-843	36	18	𝐷𝑐(a𝑢	𝐷𝑐(a𝑢	NOUN
cana-843	36	19	,	,	PUNCT
cana-843	36	20	a𝑣	a𝑣	NOUN
cana-843	36	21	)	)	PUNCT
cana-843	36	22	=	=	PRON
cana-843	36	23	{	{	PUNCT
cana-843	36	24	𝐷(a𝑢	𝐷(a𝑢	NOUN
cana-843	36	25	,	,	PUNCT
cana-843	36	26	a𝑣	a𝑣	PROPN
cana-843	36	27	)	)	PUNCT
cana-843	37	1	+	+	CCONJ
cana-843	37	2	𝑑(𝑢	𝑑(𝑢	ADJ
cana-843	37	3	,	,	PUNCT
cana-843	37	4	𝑣	𝑣	NOUN
cana-843	37	5	)	)	PUNCT
cana-843	37	6	if	if	SCONJ
cana-843	37	7	a𝑢	a𝑢	ADP
cana-843	37	8	≠	≠	PROPN
cana-843	37	9	a𝑣	a𝑣	NOUN
cana-843	37	10	0	0	PUNCT
cana-843	38	1	if	if	SCONJ
cana-843	38	2	a𝑢	a𝑢	VERB
cana-843	38	3	=	=	PUNCT
cana-843	38	4	a𝑣	a𝑣	PROPN
cana-843	38	5	the	the	DET
cana-843	38	6	detour	detour	NOUN
cana-843	38	7	distance	distance	NOUN
cana-843	38	8	and	and	CCONJ
cana-843	38	9	the	the	DET
cana-843	38	10	distance	distance	NOUN
cana-843	38	11	between	between	ADP
cana-843	38	12	𝑢	𝑢	NOUN
cana-843	38	13	and	and	CCONJ
cana-843	38	14	𝑣	𝑣	PROPN
cana-843	38	15	are	be	AUX
cana-843	38	16	denoted	denote	VERB
cana-843	38	17	by	by	ADP
cana-843	38	18	𝐷(a𝑢	𝐷(a𝑢	PROPN
cana-843	38	19	,	,	PUNCT
cana-843	38	20	𝑣	𝑣	NOUN
cana-843	38	21	)	)	PUNCT
cana-843	38	22	and	and	CCONJ
cana-843	38	23	𝑑(a𝑢	𝑑(a𝑢	NOUN
cana-843	38	24	,	,	PUNCT
cana-843	38	25	𝑣	𝑣	NOUN
cana-843	38	26	)	)	PUNCT
cana-843	38	27	,	,	PUNCT
cana-843	38	28	respectively	respectively	ADV
cana-843	38	29	.	.	PUNCT
cana-843	39	1	the	the	DET
cana-843	39	2	circular	circular	ADJ
cana-843	39	3	diameter	diameter	NOUN
cana-843	39	4	𝐷𝑐	𝐷𝑐	PROPN
cana-843	39	5	is	be	AUX
cana-843	39	6	the	the	DET
cana-843	39	7	longest	long	ADJ
cana-843	39	8	circular	circular	ADJ
cana-843	39	9	distance	distance	NOUN
cana-843	39	10	between	between	ADP
cana-843	39	11	2	2	NUM
cana-843	39	12	vertices	vertex	NOUN
cana-843	39	13	on	on	ADP
cana-843	39	14	𝐺.	𝐺.	PROPN
cana-843	39	15	an	an	DET
cana-843	39	16	𝑢	𝑢	NOUN
cana-843	39	17	−	−	NOUN
cana-843	39	18	𝑣	𝑣	DET
cana-843	39	19	circular	circular	NOUN
cana-843	39	20	of	of	ADP
cana-843	39	21	𝐺	𝐺	PROPN
cana-843	39	22	is	be	AUX
cana-843	39	23	any	any	DET
cana-843	39	24	𝑢	𝑢	NOUN
cana-843	39	25	−	−	NOUN
cana-843	39	26	𝑣	𝑣	PRON
cana-843	39	27	path	path	NOUN
cana-843	39	28	of	of	ADP
cana-843	39	29	length	length	NOUN
cana-843	39	30	𝐷𝑐(𝑢	𝐷𝑐(𝑢	NOUN
cana-843	39	31	,	,	PUNCT
cana-843	39	32	𝑣	𝑣	NOUN
cana-843	39	33	)	)	PUNCT
cana-843	39	34	.	.	PUNCT
cana-843	40	1	the	the	DET
cana-843	40	2	circular	circular	ADJ
cana-843	40	3	diameter	diameter	NOUN
cana-843	40	4	𝐷𝑐	𝐷𝑐	PROPN
cana-843	40	5	is	be	AUX
cana-843	40	6	the	the	DET
cana-843	40	7	longest	long	ADJ
cana-843	40	8	circular	circular	ADJ
cana-843	40	9	distance	distance	NOUN
cana-843	40	10	between	between	ADP
cana-843	40	11	2	2	NUM
cana-843	40	12	vertices	vertex	NOUN
cana-843	40	13	on	on	ADP
cana-843	40	14	𝐺.	𝐺.	NOUN
cana-843	40	15	for	for	ADP
cana-843	40	16	𝑢	𝑢	NOUN
cana-843	40	17	,	,	PUNCT
cana-843	40	18	𝑣	𝑣	PROPN
cana-843	40	19	∈	∈	PROPN
cana-843	40	20	𝑉	𝑉	PROPN
cana-843	40	21	,	,	PUNCT
cana-843	40	22	𝐼𝑐[𝑢	𝐼𝑐[𝑢	PROPN
cana-843	40	23	,	,	PUNCT
cana-843	40	24	𝑣	𝑣	X
cana-843	40	25	]	]	PUNCT
cana-843	40	26	represents	represent	VERB
cana-843	40	27	group	group	NOUN
cana-843	40	28	of	of	ADP
cana-843	40	29	every	every	DET
cana-843	40	30	vertex	vertex	NOUN
cana-843	40	31	positioned	position	VERB
cana-843	40	32	on	on	ADP
cana-843	40	33	a	a	DET
cana-843	40	34	𝑢	𝑢	NOUN
cana-843	40	35	−	−	NOUN
cana-843	40	36	𝑣	𝑣	DET
cana-843	40	37	circular	circular	NOUN
cana-843	40	38	in	in	ADP
cana-843	40	39	𝐺.	𝐺.	NOUN
cana-843	40	40	for	for	ADP
cana-843	40	41	𝑆	𝑆	PROPN
cana-843	40	42	⊆	⊆	NUM
cana-843	40	43	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	40	44	)	)	PUNCT
cana-843	40	45	,	,	PUNCT
cana-843	40	46	let	let	VERB
cana-843	40	47	𝐼𝑐[𝑆	𝐼𝑐[𝑆	NOUN
cana-843	40	48	]	]	X
cana-843	40	49	=	=	SYM
cana-843	40	50	⋃	⋃	PROPN
cana-843	40	51	𝐼𝑐[𝑢	𝐼𝑐[𝑢	PROPN
cana-843	40	52	,	,	PUNCT
cana-843	40	53	𝑣].𝑢,𝑣	𝑣].𝑢,𝑣	NOUN
cana-843	40	54	∈𝑆	∈𝑆	NOUN
cana-843	40	55	these	these	DET
cana-843	40	56	concepts	concept	NOUN
cana-843	40	57	were	be	AUX
cana-843	40	58	studied	study	VERB
cana-843	40	59	in	in	ADP
cana-843	40	60	[	[	X
cana-843	40	61	2,9,10	2,9,10	NUM
cana-843	40	62	]	]	X
cana-843	40	63	.	.	PUNCT
cana-843	41	1	theorem	theorem	VERB
cana-843	41	2	1.1	1.1	NUM
cana-843	41	3	.	.	PUNCT
cana-843	42	1	[	[	X
cana-843	42	2	2	2	X
cana-843	42	3	]	]	PUNCT
cana-843	42	4	in	in	ADP
cana-843	42	5	a	a	DET
cana-843	42	6	connected	connected	ADJ
cana-843	42	7	graph	graph	NOUN
cana-843	42	8	,	,	PUNCT
cana-843	42	9	every	every	DET
cana-843	42	10	geodetic	geodetic	ADJ
cana-843	42	11	set	set	NOUN
cana-843	42	12	of	of	ADP
cana-843	42	13	𝐺	𝐺	PROPN
cana-843	42	14	has	have	VERB
cana-843	42	15	an	an	DET
cana-843	42	16	extreme	extreme	ADJ
cana-843	42	17	vertex	vertex	NOUN
cana-843	42	18	.	.	PUNCT
cana-843	43	1	theorem	theorem	VERB
cana-843	43	2	1.2	1.2	NUM
cana-843	43	3	.	.	PUNCT
cana-843	44	1	[	[	X
cana-843	44	2	2	2	X
cana-843	44	3	]	]	PUNCT
cana-843	44	4	let	let	VERB
cana-843	44	5	𝑊	𝑊	PRON
cana-843	44	6	be	be	AUX
cana-843	44	7	the	the	DET
cana-843	44	8	set	set	NOUN
cana-843	44	9	of	of	ADP
cana-843	44	10	all	all	DET
cana-843	44	11	geodetic	geodetic	ADJ
cana-843	44	12	sets	set	NOUN
cana-843	44	13	in	in	ADP
cana-843	44	14	graph	graph	NOUN
cana-843	44	15	𝐺.	𝐺.	PROPN
cana-843	44	16	then	then	ADV
cana-843	44	17	𝑓𝒈(𝐺	𝑓𝒈(𝐺	CCONJ
cana-843	44	18	)	)	PUNCT
cana-843	44	19	≤	≤	NUM
cana-843	45	1	𝑔(a𝐺	𝑔(a𝐺	PROPN
cana-843	45	2	)	)	PUNCT
cana-843	45	3	–	–	PUNCT
cana-843	45	4	|a𝑊|	|a𝑊|	NOUN
cana-843	45	5	.	.	PUNCT
cana-843	46	1	2	2	NUM
cana-843	46	2	.	.	X
cana-843	46	3	the	the	DET
cana-843	46	4	forcing	force	VERB
cana-843	46	5	circular	circular	ADJ
cana-843	46	6	number	number	NOUN
cana-843	46	7	of	of	ADP
cana-843	46	8	a	a	DET
cana-843	46	9	graph	graph	NOUN
cana-843	46	10	definition	definition	NOUN
cana-843	46	11	2.1	2.1	NUM
cana-843	46	12	.	.	PUNCT
cana-843	47	1	a	a	DET
cana-843	47	2	subset	subset	NOUN
cana-843	47	3	𝑇	𝑇	PROPN
cana-843	47	4	⊆	⊆	PROPN
cana-843	47	5	𝑆	𝑆	PROPN
cana-843	47	6	is	be	AUX
cana-843	47	7	referred	refer	VERB
cana-843	47	8	to	to	ADP
cana-843	47	9	as	as	ADP
cana-843	47	10	a	a	DET
cana-843	47	11	forcing	forcing	NOUN
cana-843	47	12	subset	subset	NOUN
cana-843	47	13	for	for	ADP
cana-843	47	14	e𝑆	e𝑆	ADJ
cana-843	47	15	,	,	PUNCT
cana-843	47	16	if	if	SCONJ
cana-843	47	17	𝑆	𝑆	PROPN
cana-843	47	18	is	be	AUX
cana-843	47	19	the	the	DET
cana-843	47	20	only	only	ADJ
cana-843	47	21	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	47	22	that	that	PRON
cana-843	47	23	contains	contain	VERB
cana-843	47	24	𝑇.	𝑇.	PROPN
cana-843	47	25	a	a	DET
cana-843	47	26	forcing	force	VERB
cana-843	47	27	subset	subset	NOUN
cana-843	47	28	of	of	ADP
cana-843	47	29	minimum	minimum	ADJ
cana-843	47	30	cardinality	cardinality	NOUN
cana-843	47	31	for	for	ADP
cana-843	47	32	e𝑆	e𝑆	ADJ
cana-843	47	33	is	be	AUX
cana-843	47	34	known	know	VERB
cana-843	47	35	as	as	ADP
cana-843	47	36	a	a	DET
cana-843	47	37	minimum	minimum	ADJ
cana-843	47	38	forcing	forcing	NOUN
cana-843	47	39	subset	subset	NOUN
cana-843	47	40	of	of	ADP
cana-843	47	41	𝑆.	𝑆.	PROPN
cana-843	47	42	the	the	DET
cana-843	47	43	forcing	force	VERB
cana-843	47	44	circular	circular	ADJ
cana-843	47	45	number	number	NOUN
cana-843	47	46	of	of	ADP
cana-843	47	47	𝐺	𝐺	PROPN
cana-843	47	48	is	be	AUX
cana-843	47	49	denoted	denote	VERB
cana-843	47	50	by	by	ADP
cana-843	47	51	the	the	DET
cana-843	47	52	notation	notation	NOUN
cana-843	47	53	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	47	54	)	)	PUNCT
cana-843	47	55	=	=	PUNCT
cana-843	48	1	min{𝑓𝑐𝑟(𝑆	min{𝑓𝑐𝑟(𝑆	NOUN
cana-843	48	2	)	)	PUNCT
cana-843	48	3	}	}	PUNCT
cana-843	48	4	,	,	PUNCT
cana-843	48	5	where	where	SCONJ
cana-843	48	6	the	the	DET
cana-843	48	7	minimum	minimum	NOUN
cana-843	48	8	is	be	AUX
cana-843	48	9	established	establish	VERB
cana-843	48	10	over	over	ADP
cana-843	48	11	all	all	DET
cana-843	48	12	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	48	13	𝑆	𝑆	PROPN
cana-843	48	14	in	in	ADP
cana-843	48	15	𝐺.	𝐺.	PROPN
cana-843	49	1	the	the	DET
cana-843	49	2	cardinality	cardinality	NOUN
cana-843	49	3	of	of	ADP
cana-843	49	4	a	a	DET
cana-843	49	5	minimum	minimum	ADJ
cana-843	49	6	forcing	forcing	NOUN
cana-843	49	7	subset	subset	NOUN
cana-843	49	8	of	of	ADP
cana-843	49	9	𝑆	𝑆	PROPN
cana-843	49	10	is	be	AUX
cana-843	49	11	the	the	DET
cana-843	49	12	forcing	force	VERB
cana-843	49	13	circular	circular	ADJ
cana-843	49	14	number	number	NOUN
cana-843	49	15	”	"	PUNCT
cana-843	49	16	of	of	ADP
cana-843	49	17	𝑆.	𝑆.	PROPN
cana-843	49	18	example	example	NOUN
cana-843	49	19	2.2	2.2	NUM
cana-843	49	20	.	.	PUNCT
cana-843	50	1	the	the	DET
cana-843	50	2	only	only	ADV
cana-843	50	3	two	two	NUM
cana-843	50	4	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	50	5	of	of	ADP
cana-843	50	6	the	the	DET
cana-843	50	7	graph	graph	NOUN
cana-843	50	8	g	g	PROPN
cana-843	50	9	displayed	display	VERB
cana-843	50	10	in	in	ADP
cana-843	50	11	figure	figure	NOUN
cana-843	50	12	2.1	2.1	NUM
cana-843	50	13	are	be	AUX
cana-843	50	14	𝑆1	𝑆1	NOUN
cana-843	50	15	=	=	SYM
cana-843	50	16	{	{	PUNCT
cana-843	50	17	a𝑣1	a𝑣1	PROPN
cana-843	50	18	,	,	PUNCT
cana-843	50	19	a𝑣4	a𝑣4	PROPN
cana-843	50	20	,	,	PUNCT
cana-843	50	21	𝑣5	𝑣5	NOUN
cana-843	50	22	}	}	PUNCT
cana-843	50	23	and	and	CCONJ
cana-843	50	24	𝑆2	𝑆2	PROPN
cana-843	50	25	=	=	SYM
cana-843	50	26	{	{	PUNCT
cana-843	50	27	a𝑣1	a𝑣1	PROPN
cana-843	50	28	,	,	PUNCT
cana-843	50	29	a𝑣4	a𝑣4	PROPN
cana-843	50	30	,	,	PUNCT
cana-843	50	31	𝑣6	𝑣6	PROPN
cana-843	50	32	}	}	PUNCT
cana-843	50	33	such	such	ADJ
cana-843	50	34	that	that	SCONJ
cana-843	50	35	𝑓𝑐𝑟(𝑆1	𝑓𝑐𝑟(𝑆1	PROPN
cana-843	50	36	)	)	PUNCT
cana-843	50	37	=	=	SYM
cana-843	50	38	𝑓𝑐𝑟(𝑆2	𝑓𝑐𝑟(𝑆2	PROPN
cana-843	50	39	)	)	PUNCT
cana-843	50	40	=	=	SYM
cana-843	50	41	1	1	NUM
cana-843	50	42	and	and	CCONJ
cana-843	50	43	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	50	44	)	)	PUNCT
cana-843	50	45	=	=	SYM
cana-843	50	46	1	1	X
cana-843	50	47	.	.	X
cana-843	50	48	figure	figure	VERB
cana-843	50	49	2.1	2.1	NUM
cana-843	50	50	𝑣5	𝑣5	NOUN
cana-843	50	51	𝑣1	𝑣1	NOUN
cana-843	50	52	𝑣3	𝑣3	PROPN
cana-843	50	53	𝑣2	𝑣2	PROPN
cana-843	50	54	𝑣6	𝑣6	PROPN
cana-843	50	55	𝑣4	𝑣4	VERB
cana-843	50	56	𝐺	𝐺	NOUN
cana-843	50	57	figure	figure	VERB
cana-843	50	58	2.1	2.1	NUM
cana-843	50	59	𝑣9	𝑣9	ADJ
cana-843	50	60	𝑣7	𝑣7	ADJ
cana-843	50	61	𝑣8	𝑣8	NOUN
cana-843	50	62	communications	communication	NOUN
cana-843	50	63	on	on	ADP
cana-843	50	64	applied	apply	VERB
cana-843	50	65	nonlinear	nonlinear	ADJ
cana-843	50	66	analysis	analysis	NOUN
cana-843	50	67	issn	issn	NOUN
cana-843	50	68	:	:	PUNCT
cana-843	50	69	1074	1074	NUM
cana-843	50	70	-	-	PUNCT
cana-843	50	71	133x	133x	NUM
cana-843	50	72	vol	vol	NOUN
cana-843	50	73	31	31	NUM
cana-843	50	74	no	no	NOUN
cana-843	50	75	.	.	PUNCT
cana-843	51	1	4s	4s	NUM
cana-843	51	2	(	(	PUNCT
cana-843	51	3	2024	2024	NUM
cana-843	51	4	)	)	PUNCT
cana-843	51	5	221	221	NUM
cana-843	51	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	51	7	observation	observation	NOUN
cana-843	51	8	2.3	2.3	NUM
cana-843	51	9	.	.	PUNCT
cana-843	52	1	for	for	ADP
cana-843	52	2	each	each	DET
cana-843	52	3	graph	graph	NOUN
cana-843	52	4	𝐺	𝐺	NOUN
cana-843	52	5	that	that	PRON
cana-843	52	6	is	be	AUX
cana-843	52	7	connected	connect	VERB
cana-843	52	8	,	,	PUNCT
cana-843	52	9	0	0	NUM
cana-843	52	10	≤	≤	NUM
cana-843	52	11	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	52	12	)	)	PUNCT
cana-843	52	13	≤	≤	NOUN
cana-843	52	14	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	52	15	)	)	PUNCT
cana-843	52	16	.	.	PUNCT
cana-843	53	1	remark	remark	VERB
cana-843	53	2	2.4	2.4	NUM
cana-843	53	3	.	.	PUNCT
cana-843	54	1	observation	observation	NOUN
cana-843	54	2	2.3	2.3	NUM
cana-843	54	3	has	have	VERB
cana-843	54	4	sharp	sharp	ADJ
cana-843	54	5	bounds	bound	NOUN
cana-843	54	6	.	.	PUNCT
cana-843	55	1	for	for	ADP
cana-843	55	2	𝐺	𝐺	PROPN
cana-843	55	3	=	=	PUNCT
cana-843	55	4	𝑃3	𝑃3	NOUN
cana-843	55	5	,	,	PUNCT
cana-843	55	6	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	55	7	)	)	PUNCT
cana-843	55	8	=	=	SYM
cana-843	55	9	0	0	X
cana-843	55	10	.	.	X
cana-843	56	1	for	for	ADP
cana-843	56	2	𝐺	𝐺	PROPN
cana-843	56	3	=	=	NOUN
cana-843	56	4	𝐶4	𝐶4	NOUN
cana-843	56	5	with	with	ADP
cana-843	56	6	vertex	vertex	NOUN
cana-843	56	7	set	set	VERB
cana-843	56	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	56	9	)	)	PUNCT
cana-843	56	10	=	=	SYM
cana-843	56	11	{	{	PUNCT
cana-843	56	12	𝑣1	𝑣1	PROPN
cana-843	56	13	,	,	PUNCT
cana-843	56	14	𝑣2	𝑣2	PROPN
cana-843	56	15	,	,	PUNCT
cana-843	56	16	𝑣3	𝑣3	ADJ
cana-843	56	17	,	,	PUNCT
cana-843	56	18	𝑣4	𝑣4	NOUN
cana-843	56	19	}	}	PUNCT
cana-843	56	20	,	,	PUNCT
cana-843	56	21	𝑆1	𝑆1	NOUN
cana-843	56	22	=	=	SYM
cana-843	56	23	{	{	PUNCT
cana-843	56	24	a𝑣1	a𝑣1	PROPN
cana-843	56	25	,	,	PUNCT
cana-843	56	26	a𝑣2	a𝑣2	PROPN
cana-843	56	27	}	}	PUNCT
cana-843	56	28	,	,	PUNCT
cana-843	56	29	𝑆2	𝑆2	PROPN
cana-843	56	30	=	=	PUNCT
cana-843	56	31	{	{	PUNCT
cana-843	56	32	a𝑣2	a𝑣2	NOUN
cana-843	56	33	,	,	PUNCT
cana-843	56	34	a𝑣3	a𝑣3	NOUN
cana-843	56	35	}	}	PUNCT
cana-843	56	36	,	,	PUNCT
cana-843	56	37	𝑆3	𝑆3	PROPN
cana-843	56	38	=	=	SYM
cana-843	56	39	{	{	PUNCT
cana-843	56	40	𝑣3	𝑣3	ADJ
cana-843	56	41	,	,	PUNCT
cana-843	56	42	𝑣4	𝑣4	NOUN
cana-843	56	43	}	}	PUNCT
cana-843	56	44	,	,	PUNCT
cana-843	56	45	𝑆4	𝑆4	PROPN
cana-843	56	46	=	=	SYM
cana-843	56	47	{	{	PUNCT
cana-843	56	48	𝑣7	𝑣7	NOUN
cana-843	56	49	,	,	PUNCT
cana-843	56	50	a𝑣4	a𝑣4	PROPN
cana-843	56	51	}	}	PUNCT
cana-843	56	52	,	,	PUNCT
cana-843	56	53	𝑆5	𝑆5	PROPN
cana-843	56	54	=	=	SYM
cana-843	56	55	{	{	PUNCT
cana-843	56	56	𝑣1	𝑣1	NOUN
cana-843	56	57	,	,	PUNCT
cana-843	56	58	a𝑣3	a𝑣3	NOUN
cana-843	56	59	}	}	PUNCT
cana-843	56	60	and	and	CCONJ
cana-843	56	61	𝑆6	𝑆6	NOUN
cana-843	56	62	=	=	PUNCT
cana-843	56	63	{	{	PUNCT
cana-843	56	64	a𝑣2	a𝑣2	PROPN
cana-843	56	65	,	,	PUNCT
cana-843	56	66	a𝑣4	a𝑣4	PROPN
cana-843	56	67	}	}	PUNCT
cana-843	56	68	are	be	AUX
cana-843	56	69	the	the	DET
cana-843	56	70	only	only	ADV
cana-843	56	71	six	six	NUM
cana-843	56	72	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	56	73	of	of	ADP
cana-843	56	74	a𝐺	a𝐺	NOUN
cana-843	56	75	there	there	PRON
cana-843	56	76	exists	exist	VERB
cana-843	56	77	𝑓𝑐𝑟(𝑆𝑖	𝑓𝑐𝑟(𝑆𝑖	X
cana-843	56	78	)	)	PUNCT
cana-843	57	1	=	=	SYM
cana-843	57	2	2	2	NUM
cana-843	57	3	,	,	PUNCT
cana-843	57	4	1	1	NUM
cana-843	57	5	≤	≤	NUM
cana-843	57	6	𝑖	𝑖	SYM
cana-843	57	7	≤	≤	NUM
cana-843	57	8	6	6	NUM
cana-843	57	9	so	so	SCONJ
cana-843	57	10	that	that	SCONJ
cana-843	57	11	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	57	12	)	)	PUNCT
cana-843	57	13	=	=	SYM
cana-843	57	14	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	57	15	)	)	PUNCT
cana-843	57	16	=	=	SYM
cana-843	58	1	2	2	X
cana-843	58	2	.	.	PUNCT
cana-843	58	3	additionally	additionally	ADV
cana-843	58	4	,	,	PUNCT
cana-843	58	5	the	the	DET
cana-843	58	6	limitations	limitation	NOUN
cana-843	58	7	in	in	ADP
cana-843	58	8	observation	observation	NOUN
cana-843	58	9	2.3	2.3	NUM
cana-843	58	10	may	may	AUX
cana-843	58	11	be	be	AUX
cana-843	58	12	extremely	extremely	ADV
cana-843	58	13	rigorous	rigorous	ADJ
cana-843	58	14	.	.	PUNCT
cana-843	59	1	the	the	DET
cana-843	59	2	graph	graph	NOUN
cana-843	59	3	𝐺	𝐺	NOUN
cana-843	59	4	shown	show	VERB
cana-843	59	5	in	in	ADP
cana-843	59	6	figure	figure	NOUN
cana-843	59	7	2.1	2.1	NUM
cana-843	59	8	has	have	VERB
cana-843	59	9	two	two	NUM
cana-843	59	10	values	value	NOUN
cana-843	59	11	:	:	PUNCT
cana-843	59	12	𝑐𝑟(𝐺	𝑐𝑟(𝐺	X
cana-843	59	13	)	)	PUNCT
cana-843	59	14	=	=	SYM
cana-843	59	15	2	2	NUM
cana-843	59	16	and	and	CCONJ
cana-843	59	17	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	59	18	)	)	PUNCT
cana-843	59	19	=	=	SYM
cana-843	60	1	1	1	X
cana-843	60	2	.	.	PUNCT
cana-843	60	3	hence	hence	ADV
cana-843	60	4	0	0	NUM
cana-843	60	5	<	<	X
cana-843	60	6	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	60	7	)	)	PUNCT
cana-843	60	8	<	<	X
cana-843	60	9	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NUM
cana-843	60	10	)	)	PUNCT
cana-843	60	11	.	.	PUNCT
cana-843	61	1	theorem	theorem	VERB
cana-843	61	2	2.5	2.5	NUM
cana-843	61	3	.	.	PUNCT
cana-843	62	1	consider	consider	VERB
cana-843	62	2	a	a	DET
cana-843	62	3	connected	connected	ADJ
cana-843	62	4	graph	graph	NOUN
cana-843	62	5	,	,	PUNCT
cana-843	62	6	𝐺.	𝐺.	NOUN
cana-843	62	7	following	follow	VERB
cana-843	62	8	that	that	SCONJ
cana-843	62	9	i	i	NOUN
cana-843	62	10	)	)	PUNCT
cana-843	62	11	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	62	12	)	)	PUNCT
cana-843	62	13	=	=	SYM
cana-843	62	14	0	0	NUM
cana-843	63	1	iff	iff	PROPN
cana-843	63	2	𝐺	𝐺	PROPN
cana-843	63	3	has	have	VERB
cana-843	63	4	a	a	DET
cana-843	63	5	unique	unique	ADJ
cana-843	63	6	minimum	minimum	NOUN
cana-843	63	7	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	63	8	of	of	ADP
cana-843	63	9	𝐺.	𝐺.	PROPN
cana-843	63	10	ii	ii	NOUN
cana-843	63	11	)	)	PUNCT
cana-843	63	12	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	63	13	)	)	PUNCT
cana-843	63	14	=	=	SYM
cana-843	63	15	1	1	NUM
cana-843	63	16	iff	iff	PROPN
cana-843	63	17	𝐺	𝐺	PROPN
cana-843	63	18	possesses	possess	VERB
cana-843	63	19	a	a	DET
cana-843	63	20	minimum	minimum	NOUN
cana-843	63	21	of	of	ADP
cana-843	63	22	two	two	NUM
cana-843	63	23	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	63	24	,	,	PUNCT
cana-843	63	25	at	at	ADP
cana-843	63	26	least	least	ADJ
cana-843	63	27	one	one	NUM
cana-843	63	28	of	of	ADP
cana-843	63	29	which	which	PRON
cana-843	63	30	is	be	AUX
cana-843	63	31	a	a	DET
cana-843	63	32	distinct	distinct	NOUN
cana-843	63	33	𝑐𝑟-et	𝑐𝑟-et	NOUN
cana-843	64	1	that	that	PRON
cana-843	64	2	includes	include	VERB
cana-843	64	3	one	one	NUM
cana-843	64	4	of	of	ADP
cana-843	64	5	its	its	PRON
cana-843	64	6	elements	element	NOUN
cana-843	64	7	.	.	PUNCT
cana-843	65	1	iii	iii	X
cana-843	65	2	)	)	PUNCT
cana-843	65	3	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	65	4	)	)	PUNCT
cana-843	65	5	=	=	SYM
cana-843	65	6	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	65	7	)	)	PUNCT
cana-843	65	8	iff	iff	VERB
cana-843	65	9	any	any	DET
cana-843	65	10	proper	proper	ADJ
cana-843	65	11	subset	subset	NOUN
cana-843	65	12	of	of	ADP
cana-843	65	13	𝐺	𝐺	PROPN
cana-843	65	14	that	that	PRON
cana-843	65	15	is	be	AUX
cana-843	65	16	not	not	PART
cana-843	65	17	contained	contain	VERB
cana-843	65	18	in	in	ADP
cana-843	65	19	any	any	DET
cana-843	65	20	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	65	21	is	be	AUX
cana-843	65	22	the	the	DET
cana-843	65	23	unique	unique	ADJ
cana-843	65	24	minimal	minimal	ADJ
cana-843	65	25	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	65	26	of	of	ADP
cana-843	65	27	g.	g.	PROPN
cana-843	65	28	definition	definition	NOUN
cana-843	65	29	2.6	2.6	NUM
cana-843	65	30	.	.	PUNCT
cana-843	66	1	a	a	DET
cana-843	66	2	vertex	vertex	NOUN
cana-843	66	3	𝑣	𝑣	ADP
cana-843	66	4	“	"	PUNCT
cana-843	66	5	of	of	ADP
cana-843	66	6	a	a	DET
cana-843	66	7	connected	connected	ADJ
cana-843	66	8	graph	graph	NOUN
cana-843	66	9	𝐺.	𝐺.	NOUN
cana-843	66	10	if	if	SCONJ
cana-843	66	11	𝑣	𝑣	PRON
cana-843	66	12	belongs	belong	VERB
cana-843	66	13	to	to	ADP
cana-843	66	14	each	each	DET
cana-843	66	15	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	66	16	of	of	ADP
cana-843	66	17	𝐺	𝐺	PROPN
cana-843	66	18	,	,	PUNCT
cana-843	66	19	then	then	ADV
cana-843	66	20	𝑣(𝐺	𝑣(𝐺	PROPN
cana-843	66	21	)	)	PUNCT
cana-843	66	22	is	be	AUX
cana-843	66	23	considered	consider	VERB
cana-843	66	24	to	to	PART
cana-843	66	25	be	be	AUX
cana-843	66	26	a	a	DET
cana-843	66	27	circular	circular	ADJ
cana-843	66	28	vertex	vertex	NOUN
cana-843	66	29	of	of	ADP
cana-843	66	30	𝐺.	𝐺.	NOUN
cana-843	66	31	example	example	NOUN
cana-843	66	32	2.7	2.7	NUM
cana-843	66	33	.	.	PUNCT
cana-843	67	1	for	for	ADP
cana-843	67	2	the	the	DET
cana-843	67	3	graph	graph	NOUN
cana-843	67	4	𝐺	𝐺	NOUN
cana-843	67	5	shown	show	VERB
cana-843	67	6	in	in	ADP
cana-843	67	7	figure	figure	NOUN
cana-843	67	8	2.2	2.2	NUM
cana-843	67	9	,	,	PUNCT
cana-843	67	10	the	the	DET
cana-843	67	11	set	set	NOUN
cana-843	67	12	of	of	ADP
cana-843	67	13	all	all	DET
cana-843	67	14	circular	circular	ADJ
cana-843	67	15	vertices	vertex	NOUN
cana-843	67	16	of	of	ADP
cana-843	67	17	𝐺	𝐺	PROPN
cana-843	67	18	is	be	AUX
cana-843	67	19	represented	represent	VERB
cana-843	67	20	by	by	ADP
cana-843	67	21	{	{	PUNCT
cana-843	67	22	𝑣1	𝑣1	NOUN
cana-843	67	23	,	,	PUNCT
cana-843	67	24	𝑣3	𝑣3	ADJ
cana-843	67	25	,	,	PUNCT
cana-843	67	26	𝑣5	𝑣5	NOUN
cana-843	67	27	}	}	PUNCT
cana-843	67	28	since	since	SCONJ
cana-843	67	29	𝑆1	𝑆1	NOUN
cana-843	67	30	=	=	SYM
cana-843	67	31	{	{	PUNCT
cana-843	67	32	e𝑣1	e𝑣1	NOUN
cana-843	67	33	,	,	PUNCT
cana-843	67	34	e𝑣3	e𝑣3	PROPN
cana-843	67	35	,	,	PUNCT
cana-843	67	36	𝑣5	𝑣5	NOUN
cana-843	67	37	,	,	PUNCT
cana-843	67	38	e𝑣6	e𝑣6	NOUN
cana-843	67	39	}	}	PUNCT
cana-843	67	40	and	and	CCONJ
cana-843	67	41	𝑆2	𝑆2	PROPN
cana-843	67	42	=	=	SYM
cana-843	67	43	{	{	PUNCT
cana-843	67	44	e𝑣1	e𝑣1	NOUN
cana-843	67	45	,	,	PUNCT
cana-843	67	46	𝑣3	𝑣3	ADJ
cana-843	67	47	,	,	PUNCT
cana-843	67	48	𝑣5	𝑣5	NOUN
cana-843	67	49	,	,	PUNCT
cana-843	67	50	𝑣9	𝑣9	PROPN
cana-843	67	51	}	}	PUNCT
cana-843	67	52	are	be	AUX
cana-843	67	53	the	the	DET
cana-843	67	54	only	only	ADV
cana-843	67	55	two	two	NUM
cana-843	67	56	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	67	57	”	"	PUNCT
cana-843	67	58	of	of	ADP
cana-843	67	59	e𝐺.	e𝐺.	NOUN
cana-843	67	60	figure	figure	VERB
cana-843	67	61	2.2	2.2	NUM
cana-843	67	62	theorem	theorem	NOUN
cana-843	67	63	2.8	2.8	NUM
cana-843	67	64	.	.	PUNCT
cana-843	68	1	let	let	VERB
cana-843	68	2	𝑊	𝑊	PRON
cana-843	68	3	be	be	AUX
cana-843	68	4	the	the	DET
cana-843	68	5	set	set	NOUN
cana-843	68	6	of	of	ADP
cana-843	68	7	all	all	DET
cana-843	68	8	circular	circular	ADJ
cana-843	68	9	vertices	vertex	NOUN
cana-843	68	10	of	of	ADP
cana-843	68	11	connected	connected	ADJ
cana-843	68	12	graph	graph	NOUN
cana-843	68	13	𝐺.	𝐺.	PROPN
cana-843	68	14	then	then	ADV
cana-843	68	15	𝑓𝑐𝑟(e𝐺	𝑓𝑐𝑟(e𝐺	NOUN
cana-843	68	16	)	)	PUNCT
cana-843	68	17	≤	≤	NUM
cana-843	68	18	𝑐𝑟(e𝐺	𝑐𝑟(e𝐺	NOUN
cana-843	68	19	)	)	PUNCT
cana-843	68	20	–	–	PUNCT
cana-843	68	21	|e𝑊|	|e𝑊|	NUM
cana-843	68	22	.	.	NOUN
cana-843	68	23	remark	remark	PROPN
cana-843	68	24	2.9	2.9	NUM
cana-843	68	25	.	.	PUNCT
cana-843	69	1	the	the	DET
cana-843	69	2	bounds	bound	NOUN
cana-843	69	3	in	in	ADP
cana-843	69	4	theorem	theorem	ADJ
cana-843	69	5	2.8	2.8	NUM
cana-843	69	6	are	be	AUX
cana-843	69	7	precise	precise	ADJ
cana-843	69	8	.	.	PUNCT
cana-843	70	1	regarding	regard	VERB
cana-843	70	2	the	the	DET
cana-843	70	3	graph	graph	NOUN
cana-843	70	4	g	g	NOUN
cana-843	70	5	shown	show	VERB
cana-843	70	6	in	in	ADP
cana-843	70	7	figure	figure	NOUN
cana-843	70	8	2.2	2.2	NUM
cana-843	70	9	,	,	PUNCT
cana-843	70	10	|e𝑊|	|e𝑊|	X
cana-843	70	11	=	=	SYM
cana-843	70	12	3	3	NUM
cana-843	70	13	,	,	PUNCT
cana-843	70	14	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	70	15	)	)	PUNCT
cana-843	70	16	=	=	SYM
cana-843	70	17	4	4	NUM
cana-843	70	18	and	and	CCONJ
cana-843	70	19	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	70	20	)	)	PUNCT
cana-843	70	21	=	=	SYM
cana-843	70	22	1	1	X
cana-843	70	23	.	.	PUNCT
cana-843	70	24	thus	thus	ADV
cana-843	70	25	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	70	26	)	)	PUNCT
cana-843	70	27	=	=	SYM
cana-843	70	28	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	70	29	)	)	PUNCT
cana-843	70	30	–	–	PUNCT
cana-843	70	31	|𝑊|	|𝑊|	NOUN
cana-843	70	32	.	.	PUNCT
cana-843	71	1	moreover	moreover	ADV
cana-843	71	2	,	,	PUNCT
cana-843	71	3	the	the	DET
cana-843	71	4	bounds	bound	NOUN
cana-843	71	5	in	in	ADP
cana-843	71	6	theorem	theorem	ADJ
cana-843	71	7	2.8	2.8	NUM
cana-843	71	8	may	may	AUX
cana-843	71	9	be	be	AUX
cana-843	71	10	rigid	rigid	ADJ
cana-843	71	11	.	.	PUNCT
cana-843	72	1	with	with	ADP
cana-843	72	2	respect	respect	NOUN
cana-843	72	3	to	to	ADP
cana-843	72	4	graph	graph	NOUN
cana-843	72	5	g	g	PROPN
cana-843	72	6	displayed	display	VERB
cana-843	72	7	in	in	ADP
cana-843	72	8	figure	figure	NOUN
cana-843	72	9	2.3	2.3	NUM
cana-843	72	10	,	,	PUNCT
cana-843	72	11	𝑆1	𝑆1	NOUN
cana-843	72	12	=	=	SYM
cana-843	72	13	{	{	PUNCT
cana-843	72	14	a𝑣1	a𝑣1	PROPN
cana-843	72	15	,	,	PUNCT
cana-843	72	16	a𝑣4	a𝑣4	PROPN
cana-843	72	17	,	,	PUNCT
cana-843	72	18	a𝑣5	a𝑣5	PROPN
cana-843	72	19	,	,	PUNCT
cana-843	72	20	a𝑣7	a𝑣7	PROPN
cana-843	72	21	}	}	PUNCT
cana-843	72	22	,	,	PUNCT
cana-843	72	23	𝑆2	𝑆2	PROPN
cana-843	72	24	=	=	PUNCT
cana-843	72	25	{	{	PUNCT
cana-843	72	26	a𝑣1	a𝑣1	PROPN
cana-843	72	27	,	,	PUNCT
cana-843	72	28	a𝑣4	a𝑣4	PROPN
cana-843	72	29	,	,	PUNCT
cana-843	72	30	a𝑣5	a𝑣5	INTJ
cana-843	72	31	,	,	PUNCT
cana-843	72	32	a𝑣8	a𝑣8	ADV
cana-843	72	33	}	}	PUNCT
cana-843	72	34	,	,	PUNCT
cana-843	73	1	𝑆3	𝑆3	PROPN
cana-843	73	2	=	=	SYM
cana-843	73	3	{	{	PUNCT
cana-843	73	4	𝑣1	𝑣1	PROPN
cana-843	73	5	,	,	PUNCT
cana-843	73	6	a𝑣4	a𝑣4	PROPN
cana-843	73	7	,	,	PUNCT
cana-843	73	8	a𝑣5	a𝑣5	INTJ
cana-843	73	9	,	,	PUNCT
cana-843	73	10	𝑣9	𝑣9	ADV
cana-843	73	11	}	}	PUNCT
cana-843	73	12	,	,	PUNCT
cana-843	73	13	𝑆4	𝑆4	PROPN
cana-843	73	14	=	=	SYM
cana-843	73	15	{	{	PUNCT
cana-843	73	16	a𝑣2	a𝑣2	PROPN
cana-843	73	17	,	,	PUNCT
cana-843	73	18	a𝑣4	a𝑣4	PROPN
cana-843	73	19	,	,	PUNCT
cana-843	73	20	a𝑣5	a𝑣5	PROPN
cana-843	73	21	,	,	PUNCT
cana-843	73	22	a𝑣7	a𝑣7	PROPN
cana-843	73	23	}	}	PUNCT
cana-843	73	24	,	,	PUNCT
cana-843	73	25	𝑆5	𝑆5	PROPN
cana-843	73	26	=	=	PUNCT
cana-843	73	27	{	{	PUNCT
cana-843	73	28	a𝑣2	a𝑣2	PROPN
cana-843	73	29	,	,	PUNCT
cana-843	73	30	𝑣4	𝑣4	NOUN
cana-843	73	31	,	,	PUNCT
cana-843	73	32	a𝑣5	a𝑣5	INTJ
cana-843	73	33	,	,	PUNCT
cana-843	73	34	𝑣8	𝑣8	PROPN
cana-843	73	35	}	}	PUNCT
cana-843	73	36	and	and	CCONJ
cana-843	73	37	𝑆6	𝑆6	NOUN
cana-843	73	38	=	=	SYM
cana-843	73	39	𝑣6	𝑣6	PROPN
cana-843	73	40	𝑣1	𝑣1	PROPN
cana-843	73	41	𝑣3	𝑣3	PROPN
cana-843	73	42	𝑣2	𝑣2	PROPN
cana-843	73	43	𝑣7	𝑣7	ADJ
cana-843	73	44	𝑣4	𝑣4	NOUN
cana-843	73	45	𝑣9	𝑣9	PROPN
cana-843	73	46	𝑣10	𝑣10	VERB
cana-843	73	47	𝑣8	𝑣8	PROPN
cana-843	73	48	𝑣5	𝑣5	NOUN
cana-843	73	49	𝑣	𝑣	ADP
cana-843	73	50	𝑣13	𝑣13	PROPN
cana-843	73	51	𝑣12	𝑣12	PROPN
cana-843	73	52	𝑣11	𝑣11	VERB
cana-843	73	53	communications	communication	NOUN
cana-843	73	54	on	on	ADP
cana-843	73	55	applied	apply	VERB
cana-843	73	56	nonlinear	nonlinear	ADJ
cana-843	73	57	analysis	analysis	NOUN
cana-843	73	58	issn	issn	NOUN
cana-843	73	59	:	:	PUNCT
cana-843	73	60	1074	1074	NUM
cana-843	73	61	-	-	PUNCT
cana-843	73	62	133x	133x	NUM
cana-843	73	63	vol	vol	NOUN
cana-843	73	64	31	31	NUM
cana-843	73	65	no	no	NOUN
cana-843	73	66	.	.	PUNCT
cana-843	74	1	4s	4s	NUM
cana-843	74	2	(	(	PUNCT
cana-843	74	3	2024	2024	NUM
cana-843	74	4	)	)	PUNCT
cana-843	74	5	222	222	NUM
cana-843	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	74	7	{	{	PUNCT
cana-843	74	8	𝑣2	𝑣2	PROPN
cana-843	74	9	,	,	PUNCT
cana-843	74	10	𝑣4	𝑣4	NOUN
cana-843	74	11	,	,	PUNCT
cana-843	74	12	𝑣5	𝑣5	NOUN
cana-843	74	13	,	,	PUNCT
cana-843	74	14	𝑣9	𝑣9	PROPN
cana-843	74	15	}	}	PUNCT
cana-843	74	16	are	be	AUX
cana-843	74	17	the	the	DET
cana-843	74	18	six	six	NUM
cana-843	74	19	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	74	20	of	of	ADP
cana-843	74	21	𝐺	𝐺	NOUN
cana-843	74	22	so	so	SCONJ
cana-843	74	23	that	that	SCONJ
cana-843	74	24	{	{	PUNCT
cana-843	74	25	𝑣1	𝑣1	NOUN
cana-843	74	26	,	,	PUNCT
cana-843	74	27	𝑣4	𝑣4	NOUN
cana-843	74	28	,	,	PUNCT
cana-843	74	29	𝑣5	𝑣5	NOUN
cana-843	74	30	}	}	PUNCT
cana-843	74	31	is	be	AUX
cana-843	74	32	the	the	DET
cana-843	74	33	set	set	NOUN
cana-843	74	34	of	of	ADP
cana-843	74	35	all	all	DET
cana-843	74	36	circular	circular	ADJ
cana-843	74	37	vertices	vertex	NOUN
cana-843	74	38	of	of	ADP
cana-843	74	39	e𝐺	e𝐺	PROPN
cana-843	74	40	there	there	ADV
cana-843	74	41	exists	exist	VERB
cana-843	74	42	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	74	43	)	)	PUNCT
cana-843	74	44	=	=	SYM
cana-843	74	45	1	1	NUM
cana-843	74	46	and	and	CCONJ
cana-843	74	47	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NUM
cana-843	74	48	)	)	PUNCT
cana-843	74	49	=	=	SYM
cana-843	74	50	3	3	X
cana-843	74	51	.	.	X
cana-843	74	52	figure	figure	VERB
cana-843	74	53	2.3	2.3	NUM
cana-843	74	54	theorem	theorem	ADJ
cana-843	74	55	2.10	2.10	NUM
cana-843	74	56	.	.	PUNCT
cana-843	75	1	“	"	PUNCT
cana-843	75	2	for	for	ADP
cana-843	75	3	the	the	DET
cana-843	75	4	complete	complete	ADJ
cana-843	75	5	bipartite	bipartite	PROPN
cana-843	75	6	graph	graph	NOUN
cana-843	75	7	𝐺	𝐺	PROPN
cana-843	75	8	=	=	PUNCT
cana-843	75	9	e𝐾𝑟,𝑠	e𝐾𝑟,𝑠	PROPN
cana-843	75	10	,	,	PUNCT
cana-843	75	11	(	(	PUNCT
cana-843	75	12	1	1	NUM
cana-843	75	13	≤	≤	NOUN
cana-843	75	14	e𝑟	e𝑟	PROPN
cana-843	75	15	≤	≤	NUM
cana-843	75	16	𝑠	𝑠	PROPN
cana-843	75	17	)	)	PUNCT
cana-843	75	18	,	,	PUNCT
cana-843	75	19	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	75	20	)	)	PUNCT
cana-843	75	21	=	=	PRON
cana-843	75	22	{	{	PUNCT
cana-843	75	23	0	0	NUM
cana-843	75	24	𝑖𝑓	𝑖𝑓	NUM
cana-843	75	25	𝑟	𝑟	NOUN
cana-843	75	26	=	=	SYM
cana-843	75	27	1	1	NUM
cana-843	75	28	,	,	PUNCT
cana-843	75	29	e𝑠	e𝑠	X
cana-843	75	30	≥	≥	NOUN
cana-843	75	31	2	2	NUM
cana-843	75	32	2	2	NUM
cana-843	75	33	𝑖𝑓	𝑖𝑓	ADP
cana-843	75	34	2	2	NUM
cana-843	75	35	≤	≤	NOUN
cana-843	75	36	e𝑟	e𝑟	PROPN
cana-843	75	37	≤	≤	NUM
cana-843	75	38	e𝑠	e𝑠	ADP
cana-843	75	39	proof	proof	NOUN
cana-843	75	40	.	.	PUNCT
cana-843	76	1	let	let	VERB
cana-843	76	2	𝑈	𝑈	PROPN
cana-843	76	3	=	=	PRON
cana-843	76	4	{	{	PUNCT
cana-843	76	5	e𝑢1	e𝑢1	PROPN
cana-843	76	6	,	,	PUNCT
cana-843	76	7	e𝑢2	e𝑢2	PROPN
cana-843	76	8	,	,	PUNCT
cana-843	76	9	.	.	PUNCT
cana-843	76	10	.	.	PUNCT
cana-843	77	1	.	.	PUNCT
cana-843	78	1	,	,	PUNCT
cana-843	78	2	e𝑢𝑟	e𝑢𝑟	VERB
cana-843	78	3	}	}	PUNCT
cana-843	78	4	and	and	CCONJ
cana-843	78	5	𝑊	𝑊	PROPN
cana-843	78	6	=	=	SYM
cana-843	78	7	{	{	PUNCT
cana-843	78	8	e𝑤1	e𝑤1	PROPN
cana-843	78	9	,	,	PUNCT
cana-843	78	10	e𝑤2	e𝑤2	PROPN
cana-843	78	11	,	,	PUNCT
cana-843	78	12	.	.	PUNCT
cana-843	78	13	.	.	PUNCT
cana-843	79	1	.	.	PUNCT
cana-843	80	1	,	,	PUNCT
cana-843	80	2	e𝑤𝑠	e𝑤𝑠	NOUN
cana-843	80	3	}	}	PUNCT
cana-843	80	4	be	be	VERB
cana-843	80	5	the	the	DET
cana-843	80	6	bipartite	bipartite	PROPN
cana-843	80	7	sets	set	NOUN
cana-843	80	8	of	of	ADP
cana-843	80	9	𝐺.	𝐺.	NOUN
cana-843	80	10	for	for	ADP
cana-843	80	11	𝑠	𝑠	PROPN
cana-843	80	12	≥	≥	NUM
cana-843	80	13	2	2	NUM
cana-843	80	14	and	and	CCONJ
cana-843	80	15	𝑟	𝑟	NOUN
cana-843	80	16	=	=	SYM
cana-843	80	17	1	1	NUM
cana-843	80	18	,	,	PUNCT
cana-843	80	19	𝑆	𝑆	PROPN
cana-843	80	20	=	=	SYM
cana-843	80	21	𝑊	𝑊	PROPN
cana-843	80	22	is	be	AUX
cana-843	80	23	the	the	DET
cana-843	80	24	distinct	distinct	ADJ
cana-843	80	25	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	80	26	of	of	ADP
cana-843	80	27	e𝐺	e𝐺	PROPN
cana-843	80	28	so	so	SCONJ
cana-843	80	29	that	that	SCONJ
cana-843	80	30	𝑓𝑐𝑟(e𝐺	𝑓𝑐𝑟(e𝐺	X
cana-843	80	31	)	)	PUNCT
cana-843	81	1	=	=	SYM
cana-843	81	2	0	0	X
cana-843	81	3	.	.	PUNCT
cana-843	82	1	hence	hence	ADV
cana-843	82	2	2	2	NUM
cana-843	82	3	≤	≤	NOUN
cana-843	82	4	e𝑟	e𝑟	PRON
cana-843	82	5	≤	≤	PROPN
cana-843	82	6	e𝑠.	e𝑠.	ADV
cana-843	82	7	let	let	VERB
cana-843	82	8	𝑤	𝑤	ADP
cana-843	82	9	∈	∈	PROPN
cana-843	82	10	𝑊	𝑊	PROPN
cana-843	82	11	and	and	CCONJ
cana-843	82	12	𝑢	𝑢	PRON
cana-843	82	13	∈	∈	PROPN
cana-843	82	14	𝑈.	𝑈.	NOUN
cana-843	82	15	such	such	ADJ
cana-843	82	16	that	that	PRON
cana-843	82	17	𝑆	𝑆	PROPN
cana-843	82	18	=	=	PRON
cana-843	82	19	{	{	PUNCT
cana-843	82	20	𝑢,𝑤	𝑢,𝑤	NOUN
cana-843	82	21	}	}	PUNCT
cana-843	82	22	is	be	AUX
cana-843	82	23	a	a	DET
cana-843	82	24	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	82	25	of	of	ADP
cana-843	82	26	𝐺.	𝐺.	NOUN
cana-843	82	27	”	"	PUNCT
cana-843	82	28	since	since	SCONJ
cana-843	82	29	this	this	PRON
cana-843	82	30	is	be	AUX
cana-843	82	31	true	true	ADJ
cana-843	82	32	for	for	ADP
cana-843	82	33	all	all	DET
cana-843	82	34	e𝑢∈	e𝑢∈	PROPN
cana-843	82	35	e𝑈	e𝑈	NOUN
cana-843	82	36	and	and	CCONJ
cana-843	82	37	e𝑤∈	e𝑤∈	PROPN
cana-843	82	38	e𝑊	e𝑊	PROPN
cana-843	82	39	,	,	PUNCT
cana-843	82	40	𝑆	𝑆	PROPN
cana-843	82	41	is	be	AUX
cana-843	82	42	not	not	PART
cana-843	82	43	unique	unique	ADJ
cana-843	82	44	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	82	45	of	of	ADP
cana-843	82	46	𝐺	𝐺	PROPN
cana-843	82	47	containing	contain	VERB
cana-843	82	48	𝑢	𝑢	NOUN
cana-843	82	49	or	or	CCONJ
cana-843	82	50	𝑤.	𝑤.	NOUN
cana-843	82	51	therefore	therefore	ADV
cana-843	82	52	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	82	53	)	)	PUNCT
cana-843	82	54	=	=	SYM
cana-843	83	1	2	2	X
cana-843	83	2	.	.	PUNCT
cana-843	84	1	as	as	SCONJ
cana-843	84	2	this	this	PRON
cana-843	84	3	holds	hold	VERB
cana-843	84	4	“	"	PUNCT
cana-843	84	5	true	true	ADJ
cana-843	84	6	for	for	ADP
cana-843	84	7	every	every	DET
cana-843	84	8	𝑐𝑟sets	𝑐𝑟set	NOUN
cana-843	84	9	𝑆	𝑆	PROPN
cana-843	84	10	of	of	ADP
cana-843	84	11	e𝐺	e𝐺	PROPN
cana-843	84	12	,	,	PUNCT
cana-843	84	13	𝑓𝑐𝑟(e𝐺	𝑓𝑐𝑟(e𝐺	NUM
cana-843	84	14	)	)	PUNCT
cana-843	84	15	=	=	SYM
cana-843	84	16	2	2	X
cana-843	84	17	.	.	X
cana-843	84	18	theorem	theorem	VERB
cana-843	84	19	2.11	2.11	NUM
cana-843	84	20	.	.	PUNCT
cana-843	85	1	for	for	ADP
cana-843	85	2	the	the	DET
cana-843	85	3	non	non	ADJ
cana-843	85	4	-	-	ADJ
cana-843	85	5	trivial	trivial	ADJ
cana-843	85	6	tree	tree	NOUN
cana-843	85	7	𝑇	𝑇	PROPN
cana-843	85	8	,	,	PUNCT
cana-843	85	9	𝑓𝑐𝑟(𝑇	𝑓𝑐𝑟(𝑇	NUM
cana-843	85	10	)	)	PUNCT
cana-843	86	1	=	=	SYM
cana-843	86	2	0	0	X
cana-843	86	3	.	.	PUNCT
cana-843	87	1	proof	proof	NOUN
cana-843	87	2	.	.	PUNCT
cana-843	88	1	considering	consider	VERB
cana-843	88	2	𝑆	𝑆	PROPN
cana-843	88	3	to	to	PART
cana-843	88	4	be	be	AUX
cana-843	88	5	the	the	DET
cana-843	88	6	collection	collection	NOUN
cana-843	88	7	of	of	ADP
cana-843	88	8	all	all	DET
cana-843	88	9	end	end	NOUN
cana-843	88	10	vertices	vertex	NOUN
cana-843	88	11	in	in	ADP
cana-843	88	12	𝐺	𝐺	PROPN
cana-843	88	13	,	,	PUNCT
cana-843	88	14	𝑆	𝑆	PROPN
cana-843	88	15	is	be	AUX
cana-843	88	16	the	the	DET
cana-843	88	17	only	only	ADJ
cana-843	88	18	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	88	19	in	in	ADP
cana-843	88	20	𝐺	𝐺	PROPN
cana-843	89	1	such	such	ADJ
cana-843	89	2	that	that	PRON
cana-843	89	3	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	89	4	)	)	PUNCT
cana-843	89	5	=	=	SYM
cana-843	89	6	0	0	X
cana-843	89	7	.	.	PUNCT
cana-843	89	8	theorem	theorem	VERB
cana-843	89	9	2.12	2.12	NUM
cana-843	89	10	.	.	PUNCT
cana-843	90	1	“	"	PUNCT
cana-843	90	2	for	for	ADP
cana-843	90	3	the	the	DET
cana-843	90	4	cycle	cycle	NOUN
cana-843	90	5	e𝐺	e𝐺	PROPN
cana-843	90	6	=	=	PUNCT
cana-843	90	7	e𝐶𝑛,(e𝑛≥4	e𝐶𝑛,(e𝑛≥4	NUM
cana-843	90	8	)	)	PUNCT
cana-843	90	9	,	,	PUNCT
cana-843	90	10	𝑓𝑐𝑟(e𝐺	𝑓𝑐𝑟(e𝐺	NUM
cana-843	90	11	)	)	PUNCT
cana-843	90	12	=	=	SYM
cana-843	90	13	2	2	X
cana-843	90	14	.	.	PUNCT
cana-843	90	15	”	"	PUNCT
cana-843	91	1	proof	proof	NOUN
cana-843	91	2	.	.	PUNCT
cana-843	92	1	let	let	VERB
cana-843	92	2	𝑥	𝑥	NOUN
cana-843	92	3	and	and	CCONJ
cana-843	92	4	𝑦	𝑦	PRON
cana-843	92	5	represent	represent	VERB
cana-843	92	6	any	any	DET
cana-843	92	7	two	two	NUM
cana-843	92	8	vertices	vertex	NOUN
cana-843	92	9	of	of	ADP
cana-843	92	10	𝐺.	𝐺.	NOUN
cana-843	92	11	there	there	ADV
cana-843	92	12	exists	exist	VERB
cana-843	92	13	𝑆	𝑆	PROPN
cana-843	92	14	=	=	SYM
cana-843	92	15	{	{	PUNCT
cana-843	92	16	𝑥	𝑥	PROPN
cana-843	92	17	,	,	PUNCT
cana-843	92	18	𝑦	𝑦	NOUN
cana-843	92	19	}	}	PUNCT
cana-843	92	20	is	be	AUX
cana-843	92	21	a	a	DET
cana-843	92	22	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	92	23	of	of	ADP
cana-843	92	24	𝐺.	𝐺.	NOUN
cana-843	92	25	hence	hence	ADV
cana-843	92	26	e𝑥	e𝑥	NOUN
cana-843	92	27	and	and	CCONJ
cana-843	92	28	e𝑦	e𝑦	PROPN
cana-843	92	29	are	be	AUX
cana-843	92	30	arbitrary	arbitrary	ADJ
cana-843	92	31	,	,	PUNCT
cana-843	92	32	𝑆	𝑆	PROPN
cana-843	92	33	is	be	AUX
cana-843	92	34	not	not	PART
cana-843	92	35	a	a	DET
cana-843	92	36	unique	unique	ADJ
cana-843	92	37	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	92	38	containing	contain	VERB
cana-843	92	39	𝑥	𝑥	PRON
cana-843	92	40	or	or	CCONJ
cana-843	92	41	𝑦.	𝑦.	X
cana-843	92	42	therefore	therefore	ADV
cana-843	92	43	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	92	44	)	)	PUNCT
cana-843	92	45	=	=	SYM
cana-843	93	1	2	2	X
cana-843	93	2	.	.	PUNCT
cana-843	93	3	as	as	SCONJ
cana-843	93	4	this	this	PRON
cana-843	93	5	holds	hold	VERB
cana-843	93	6	true	true	ADJ
cana-843	93	7	for	for	ADP
cana-843	93	8	all	all	DET
cana-843	93	9	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	93	10	𝑆	𝑆	PROPN
cana-843	93	11	of	of	ADP
cana-843	93	12	𝐺	𝐺	PROPN
cana-843	93	13	”	"	PUNCT
cana-843	93	14	therefore	therefore	ADV
cana-843	93	15	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	93	16	)	)	PUNCT
cana-843	93	17	=	=	SYM
cana-843	93	18	2	2	X
cana-843	93	19	.	.	PUNCT
cana-843	93	20	theorem	theorem	NOUN
cana-843	93	21	2.13	2.13	NUM
cana-843	93	22	.	.	PUNCT
cana-843	94	1	“	"	PUNCT
cana-843	94	2	for	for	ADP
cana-843	94	3	the	the	DET
cana-843	94	4	wheel	wheel	NOUN
cana-843	94	5	e𝐺	e𝐺	PROPN
cana-843	94	6	=	=	PUNCT
cana-843	94	7	e𝐾	e𝐾	NOUN
cana-843	94	8	1	1	NUM
cana-843	94	9	+	+	ADJ
cana-843	94	10	𝐶	𝐶	PROPN
cana-843	94	11	𝑛−1	𝑛−1	PROPN
cana-843	94	12	,	,	PUNCT
cana-843	94	13	(	(	PUNCT
cana-843	94	14	e𝑛≥5	e𝑛≥5	PROPN
cana-843	94	15	)	)	PUNCT
cana-843	94	16	,	,	PUNCT
cana-843	94	17	𝑓𝑐𝑟(e𝐺	𝑓𝑐𝑟(e𝐺	NUM
cana-843	94	18	)	)	PUNCT
cana-843	94	19	=	=	SYM
cana-843	94	20	1	1	X
cana-843	94	21	.	.	PUNCT
cana-843	94	22	”	"	PUNCT
cana-843	94	23	proof	proof	NOUN
cana-843	94	24	.	.	PUNCT
cana-843	95	1	assume	assume	VERB
cana-843	95	2	that	that	SCONJ
cana-843	95	3	𝑥	𝑥	PROPN
cana-843	95	4	represents	represent	VERB
cana-843	95	5	the	the	DET
cana-843	95	6	central	central	ADJ
cana-843	95	7	vertex	vertex	NOUN
cana-843	95	8	of	of	ADP
cana-843	95	9	𝐺	𝐺	PROPN
cana-843	95	10	and	and	CCONJ
cana-843	95	11	e𝐶𝑛−1	e𝐶𝑛−1	NOUN
cana-843	95	12	be	be	AUX
cana-843	95	13	𝑣1	𝑣1	PROPN
cana-843	95	14	,	,	PUNCT
cana-843	95	15	𝑣2	𝑣2	PROPN
cana-843	95	16	,	,	PUNCT
cana-843	95	17	…	…	PUNCT
cana-843	95	18	,	,	PUNCT
cana-843	95	19	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-843	95	20	,	,	PUNCT
cana-843	95	21	𝑣1	𝑣1	PROPN
cana-843	95	22	.	.	PUNCT
cana-843	96	1	then	then	ADV
cana-843	96	2	𝑆𝑖	𝑆𝑖	PROPN
cana-843	96	3	=	=	PUNCT
cana-843	96	4	{	{	PUNCT
cana-843	96	5	𝑥	𝑥	NOUN
cana-843	96	6	,	,	PUNCT
cana-843	96	7	𝑣𝑖	𝑣𝑖	ADP
cana-843	96	8	}	}	PUNCT
cana-843	96	9	(	(	PUNCT
cana-843	96	10	1	1	NUM
cana-843	96	11	≤	≤	NUM
cana-843	96	12	𝑖	𝑖	SYM
cana-843	96	13	≤	≤	NUM
cana-843	97	1	𝑛	𝑛	PRON
cana-843	97	2	−	−	PROPN
cana-843	97	3	1	1	NUM
cana-843	97	4	)	)	PUNCT
cana-843	97	5	and	and	CCONJ
cana-843	97	6	𝑆	𝑆	PROPN
cana-843	97	7	=	=	SYM
cana-843	97	8	{	{	PUNCT
cana-843	97	9	𝑢	𝑢	X
cana-843	97	10	,	,	PUNCT
cana-843	97	11	𝑣	𝑣	NOUN
cana-843	97	12	}	}	PUNCT
cana-843	97	13	where	where	SCONJ
cana-843	97	14	𝑢	𝑢	NOUN
cana-843	97	15	and	and	CCONJ
cana-843	97	16	𝑣	𝑣	PROPN
cana-843	97	17	are	be	AUX
cana-843	97	18	any	any	DET
cana-843	97	19	two	two	NUM
cana-843	97	20	vertices	vertex	NOUN
cana-843	97	21	in	in	ADP
cana-843	97	22	𝐶𝑛−1	𝐶𝑛−1	NOUN
cana-843	97	23	are	be	AUX
cana-843	97	24	the	the	DET
cana-843	97	25	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	97	26	of	of	ADP
cana-843	97	27	𝐺.	𝐺.	NOUN
cana-843	97	28	now	now	ADV
cana-843	97	29	𝑓𝑐𝑟(𝑆𝑖	𝑓𝑐𝑟(𝑆𝑖	X
cana-843	97	30	)	)	PUNCT
cana-843	98	1	=	=	SYM
cana-843	98	2	1	1	NUM
cana-843	98	3	(	(	PUNCT
cana-843	98	4	1	1	NUM
cana-843	98	5	≤	≤	NUM
cana-843	98	6	𝑖	𝑖	SYM
cana-843	98	7	≤	≤	NUM
cana-843	98	8	𝑛	𝑛	PRON
cana-843	98	9	−	−	PROPN
cana-843	98	10	1	1	NUM
cana-843	98	11	)	)	PUNCT
cana-843	98	12	.	.	PUNCT
cana-843	99	1	since	since	SCONJ
cana-843	99	2	𝑢	𝑢	PROPN
cana-843	99	3	and	and	CCONJ
cana-843	99	4	𝑣	𝑣	PRON
cana-843	99	5	are	be	AUX
cana-843	99	6	arbitrary	arbitrary	ADJ
cana-843	99	7	,	,	PUNCT
cana-843	99	8	𝑆	𝑆	PROPN
cana-843	99	9	is	be	AUX
cana-843	99	10	not	not	PART
cana-843	99	11	a	a	DET
cana-843	99	12	dsitinct	dsitinct	ADJ
cana-843	99	13	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	99	14	containing	contain	VERB
cana-843	99	15	𝑢	𝑢	NOUN
cana-843	99	16	or	or	CCONJ
cana-843	99	17	𝑣.	𝑣.	NOUN
cana-843	99	18	therefore	therefore	ADV
cana-843	99	19	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	99	20	)	)	PUNCT
cana-843	99	21	=	=	SYM
cana-843	100	1	2	2	X
cana-843	100	2	.	.	X
cana-843	100	3	hence	hence	ADV
cana-843	100	4	it	it	PRON
cana-843	100	5	follows	follow	VERB
cana-843	100	6	that	that	SCONJ
cana-843	100	7	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	100	8	)	)	PUNCT
cana-843	100	9	=	=	SYM
cana-843	100	10	1	1	X
cana-843	100	11	.	.	PUNCT
cana-843	100	12	theorem	theorem	VERB
cana-843	100	13	2.14	2.14	NUM
cana-843	100	14	.	.	PUNCT
cana-843	101	1	“	"	PUNCT
cana-843	101	2	for	for	ADP
cana-843	101	3	the	the	DET
cana-843	101	4	fan	fan	NOUN
cana-843	101	5	graph	graph	NOUN
cana-843	101	6	𝐹𝑛	𝐹𝑛	PROPN
cana-843	101	7	=	=	SYM
cana-843	101	8	e𝐾1	e𝐾1	PROPN
cana-843	101	9	+	+	PROPN
cana-843	101	10	𝑃𝑛−1	𝑃𝑛−1	PROPN
cana-843	101	11	,	,	PUNCT
cana-843	101	12	(	(	PUNCT
cana-843	101	13	e𝑛	e𝑛	X
cana-843	101	14	≥	≥	NUM
cana-843	101	15	5	5	NUM
cana-843	101	16	)	)	PUNCT
cana-843	101	17	,	,	PUNCT
cana-843	101	18	𝑓𝑐𝑟(e𝐺	𝑓𝑐𝑟(e𝐺	NUM
cana-843	101	19	)	)	PUNCT
cana-843	101	20	=	=	SYM
cana-843	101	21	1	1	X
cana-843	101	22	.	.	PUNCT
cana-843	101	23	”	"	PUNCT
cana-843	101	24	communications	communication	NOUN
cana-843	101	25	on	on	ADP
cana-843	101	26	applied	apply	VERB
cana-843	101	27	nonlinear	nonlinear	ADJ
cana-843	101	28	analysis	analysis	NOUN
cana-843	101	29	issn	issn	NOUN
cana-843	101	30	:	:	PUNCT
cana-843	101	31	1074	1074	NUM
cana-843	101	32	-	-	PUNCT
cana-843	101	33	133x	133x	NUM
cana-843	101	34	vol	vol	NOUN
cana-843	101	35	31	31	NUM
cana-843	101	36	no	no	NOUN
cana-843	101	37	.	.	PUNCT
cana-843	102	1	4s	4s	NUM
cana-843	102	2	(	(	PUNCT
cana-843	102	3	2024	2024	NUM
cana-843	102	4	)	)	PUNCT
cana-843	102	5	223	223	NUM
cana-843	102	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	102	7	proof	proof	NOUN
cana-843	102	8	.	.	PUNCT
cana-843	103	1	suppose	suppose	VERB
cana-843	103	2	that	that	SCONJ
cana-843	103	3	𝑥	𝑥	PROPN
cana-843	103	4	represents	represent	VERB
cana-843	103	5	the	the	DET
cana-843	103	6	central	central	ADJ
cana-843	103	7	vertex	vertex	NOUN
cana-843	103	8	of	of	ADP
cana-843	103	9	of	of	ADP
cana-843	103	10	𝐺	𝐺	PROPN
cana-843	103	11	and	and	CCONJ
cana-843	103	12	𝑉(𝑃𝑛−1	𝑉(𝑃𝑛−1	NOUN
cana-843	103	13	)	)	PUNCT
cana-843	104	1	=	=	PRON
cana-843	104	2	{	{	PUNCT
cana-843	104	3	𝑣1	𝑣1	PROPN
cana-843	104	4	,	,	PUNCT
cana-843	104	5	𝑣2	𝑣2	PROPN
cana-843	104	6	,	,	PUNCT
cana-843	104	7	…	…	PUNCT
cana-843	104	8	,	,	PUNCT
cana-843	104	9	𝑣𝑛−1	𝑣𝑛−1	PROPN
cana-843	104	10	}	}	PUNCT
cana-843	104	11	.	.	PUNCT
cana-843	105	1	then	then	ADV
cana-843	105	2	𝑆𝑖	𝑆𝑖	PROPN
cana-843	105	3	=	=	PUNCT
cana-843	105	4	{	{	PUNCT
cana-843	105	5	𝑥	𝑥	NOUN
cana-843	105	6	,	,	PUNCT
cana-843	105	7	𝑣𝑖	𝑣𝑖	ADP
cana-843	105	8	}	}	PUNCT
cana-843	105	9	(	(	PUNCT
cana-843	105	10	1	1	NUM
cana-843	105	11	≤	≤	NUM
cana-843	105	12	𝑖	𝑖	SYM
cana-843	105	13	≤	≤	NUM
cana-843	105	14	𝑛	𝑛	PRON
cana-843	105	15	−	−	PROPN
cana-843	105	16	1	1	NUM
cana-843	105	17	)	)	PUNCT
cana-843	105	18	and	and	CCONJ
cana-843	105	19	𝑆	𝑆	PROPN
cana-843	105	20	=	=	SYM
cana-843	105	21	{	{	PUNCT
cana-843	105	22	𝑢	𝑢	X
cana-843	105	23	,	,	PUNCT
cana-843	105	24	𝑣	𝑣	NOUN
cana-843	105	25	}	}	PUNCT
cana-843	105	26	where	where	SCONJ
cana-843	105	27	𝑢	𝑢	NOUN
cana-843	105	28	and	and	CCONJ
cana-843	105	29	𝑣	𝑣	PROPN
cana-843	105	30	are	be	AUX
cana-843	105	31	any	any	DET
cana-843	105	32	two	two	NUM
cana-843	105	33	vertices	vertex	NOUN
cana-843	105	34	in	in	ADP
cana-843	105	35	𝑃𝑛−1	𝑃𝑛−1	PROPN
cana-843	105	36	are	be	AUX
cana-843	105	37	the	the	DET
cana-843	105	38	𝑐𝑟sets	𝑐𝑟set	NOUN
cana-843	105	39	of	of	ADP
cana-843	105	40	𝐺.	𝐺.	NOUN
cana-843	105	41	now	now	ADV
cana-843	105	42	𝑓𝑐𝑟(𝑆𝑖	𝑓𝑐𝑟(𝑆𝑖	X
cana-843	105	43	)	)	PUNCT
cana-843	106	1	=	=	SYM
cana-843	106	2	1	1	NUM
cana-843	106	3	(	(	PUNCT
cana-843	106	4	1	1	NUM
cana-843	106	5	≤	≤	NUM
cana-843	106	6	𝑖	𝑖	SYM
cana-843	106	7	≤	≤	NUM
cana-843	106	8	𝑛	𝑛	PRON
cana-843	106	9	−	−	PROPN
cana-843	106	10	1	1	NUM
cana-843	106	11	)	)	PUNCT
cana-843	106	12	.	.	PUNCT
cana-843	107	1	since	since	SCONJ
cana-843	107	2	𝑢	𝑢	PROPN
cana-843	107	3	and	and	CCONJ
cana-843	107	4	𝑣	𝑣	PRON
cana-843	107	5	are	be	AUX
cana-843	107	6	arbitrary	arbitrary	ADJ
cana-843	107	7	,	,	PUNCT
cana-843	107	8	𝑆	𝑆	PROPN
cana-843	107	9	is	be	AUX
cana-843	107	10	not	not	PART
cana-843	107	11	a	a	DET
cana-843	107	12	unique	unique	ADJ
cana-843	107	13	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	107	14	containing	contain	VERB
cana-843	107	15	𝑢	𝑢	NOUN
cana-843	107	16	or	or	CCONJ
cana-843	107	17	𝑣.	𝑣.	NOUN
cana-843	107	18	therefore	therefore	ADV
cana-843	107	19	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	107	20	)	)	PUNCT
cana-843	107	21	=	=	SYM
cana-843	108	1	2	2	X
cana-843	108	2	.	.	X
cana-843	108	3	hence	hence	ADV
cana-843	108	4	it	it	PRON
cana-843	108	5	follows	follow	VERB
cana-843	108	6	that	that	SCONJ
cana-843	108	7	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	108	8	)	)	PUNCT
cana-843	108	9	=	=	SYM
cana-843	109	1	1	1	NUM
cana-843	109	2	.	.	NOUN
cana-843	109	3	3	3	NUM
cana-843	109	4	.	.	X
cana-843	110	1	the	the	DET
cana-843	110	2	forcing	force	VERB
cana-843	110	3	geodetic	geodetic	ADJ
cana-843	110	4	numbers	number	NOUN
cana-843	110	5	and	and	CCONJ
cana-843	110	6	the	the	DET
cana-843	110	7	forcing	force	VERB
cana-843	110	8	circular	circular	ADJ
cana-843	110	9	number	number	NOUN
cana-843	110	10	of	of	ADP
cana-843	110	11	a	a	DET
cana-843	110	12	graph	graph	NOUN
cana-843	110	13	the	the	DET
cana-843	110	14	forcing	force	VERB
cana-843	110	15	geodetic	geodetic	ADJ
cana-843	110	16	numbers	number	NOUN
cana-843	110	17	and	and	CCONJ
cana-843	110	18	the	the	DET
cana-843	110	19	forcing	force	VERB
cana-843	110	20	circular	circular	ADJ
cana-843	110	21	number	number	NOUN
cana-843	110	22	of	of	ADP
cana-843	110	23	a	a	DET
cana-843	110	24	graph	graph	NOUN
cana-843	110	25	have	have	VERB
cana-843	110	26	no	no	DET
cana-843	110	27	relationship	relationship	NOUN
cana-843	110	28	,	,	PUNCT
cana-843	110	29	as	as	ADP
cana-843	110	30	the	the	DET
cana-843	110	31	example	example	NOUN
cana-843	110	32	below	below	ADP
cana-843	110	33	demonstrates	demonstrate	NOUN
cana-843	110	34	.	.	PUNCT
cana-843	110	35	example	example	NOUN
cana-843	111	1	3.1	3.1	NUM
cana-843	111	2	.	.	PUNCT
cana-843	112	1	the	the	DET
cana-843	112	2	unique	unique	ADJ
cana-843	112	3	𝑔-set	𝑔-set	NOUN
cana-843	112	4	of	of	ADP
cana-843	112	5	the	the	DET
cana-843	112	6	graph	graph	NOUN
cana-843	112	7	𝐺	𝐺	NOUN
cana-843	112	8	shown	show	VERB
cana-843	112	9	in	in	ADP
cana-843	112	10	figure	figure	NOUN
cana-843	112	11	3.1	3.1	NUM
cana-843	112	12	is	be	AUX
cana-843	112	13	indicated	indicate	VERB
cana-843	112	14	as	as	ADP
cana-843	112	15	,	,	PUNCT
cana-843	112	16	𝑆	𝑆	PROPN
cana-843	112	17	=	=	SYM
cana-843	112	18	{	{	PUNCT
cana-843	112	19	𝑣1	𝑣1	PROPN
cana-843	112	20	,	,	PUNCT
cana-843	112	21	𝑣4	𝑣4	NOUN
cana-843	112	22	,	,	PUNCT
cana-843	112	23	𝑣5	𝑣5	NOUN
cana-843	112	24	}	}	PUNCT
cana-843	112	25	.	.	PUNCT
cana-843	113	1	therefore	therefore	ADV
cana-843	113	2	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	113	3	)	)	PUNCT
cana-843	113	4	=	=	SYM
cana-843	114	1	0	0	X
cana-843	114	2	.	.	PUNCT
cana-843	114	3	also	also	ADV
cana-843	114	4	“	"	PUNCT
cana-843	114	5	𝑆1	𝑆1	NOUN
cana-843	114	6	=	=	SYM
cana-843	114	7	{	{	PUNCT
cana-843	114	8	𝑣1	𝑣1	NOUN
cana-843	114	9	,	,	PUNCT
cana-843	114	10	𝑣4	𝑣4	NOUN
cana-843	114	11	}	}	PUNCT
cana-843	114	12	and	and	CCONJ
cana-843	114	13	𝑆2	𝑆2	PROPN
cana-843	114	14	=	=	SYM
cana-843	114	15	{	{	PUNCT
cana-843	114	16	𝑣1	𝑣1	PROPN
cana-843	114	17	,	,	PUNCT
cana-843	114	18	𝑣5	𝑣5	NOUN
cana-843	114	19	}	}	PUNCT
cana-843	114	20	are	be	AUX
cana-843	114	21	the	the	DET
cana-843	114	22	only	only	ADV
cana-843	114	23	two	two	NUM
cana-843	114	24	𝑐𝑟-sets	𝑐𝑟-set	NOUN
cana-843	114	25	of	of	ADP
cana-843	114	26	𝐺	𝐺	PROPN
cana-843	114	27	such	such	ADJ
cana-843	114	28	that	that	PRON
cana-843	114	29	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	114	30	)	)	PUNCT
cana-843	114	31	=	=	SYM
cana-843	115	1	1	1	X
cana-843	115	2	.	.	PUNCT
cana-843	115	3	thus	thus	ADV
cana-843	115	4	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	115	5	)	)	PUNCT
cana-843	115	6	<	<	X
cana-843	115	7	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	115	8	)	)	PUNCT
cana-843	115	9	.	.	PUNCT
cana-843	116	1	example	example	NOUN
cana-843	116	2	3.2	3.2	NUM
cana-843	116	3	.	.	PUNCT
cana-843	117	1	the	the	DET
cana-843	117	2	unique	unique	ADJ
cana-843	117	3	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	117	4	of	of	ADP
cana-843	117	5	the	the	DET
cana-843	117	6	graph	graph	NOUN
cana-843	117	7	𝐺	𝐺	NOUN
cana-843	117	8	shown	show	VERB
cana-843	117	9	in	in	ADP
cana-843	117	10	figure	figure	NOUN
cana-843	117	11	3.2	3.2	NUM
cana-843	117	12	is	be	AUX
cana-843	117	13	represented	represent	VERB
cana-843	117	14	as	as	ADP
cana-843	117	15	𝑆	𝑆	PROPN
cana-843	117	16	=	=	SYM
cana-843	117	17	{	{	PUNCT
cana-843	117	18	𝑣1	𝑣1	PROPN
cana-843	117	19	,	,	PUNCT
cana-843	117	20	𝑣2	𝑣2	PROPN
cana-843	117	21	}	}	PUNCT
cana-843	117	22	.	.	PUNCT
cana-843	118	1	therefore	therefore	ADV
cana-843	118	2	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	118	3	)	)	PUNCT
cana-843	118	4	=	=	SYM
cana-843	119	1	0	0	X
cana-843	119	2	.	.	PUNCT
cana-843	119	3	also	also	ADV
cana-843	119	4	𝑆1	𝑆1	NOUN
cana-843	119	5	=	=	SYM
cana-843	119	6	{	{	PUNCT
cana-843	119	7	𝑣1	𝑣1	NOUN
cana-843	119	8	,	,	PUNCT
cana-843	119	9	𝑣3	𝑣3	ADJ
cana-843	119	10	,	,	PUNCT
cana-843	119	11	𝑣6	𝑣6	PROPN
cana-843	119	12	}	}	PUNCT
cana-843	119	13	and	and	CCONJ
cana-843	119	14	𝑆2	𝑆2	PROPN
cana-843	119	15	=	=	SYM
cana-843	119	16	{	{	PUNCT
cana-843	119	17	𝑣1	𝑣1	PROPN
cana-843	119	18	,	,	PUNCT
cana-843	119	19	𝑣4	𝑣4	NOUN
cana-843	119	20	,	,	PUNCT
cana-843	119	21	𝑣6	𝑣6	PROPN
cana-843	119	22	}	}	PUNCT
cana-843	119	23	are	be	AUX
cana-843	119	24	the	the	DET
cana-843	119	25	only	only	ADJ
cana-843	119	26	𝑔-sets	𝑔-set	NOUN
cana-843	119	27	of	of	ADP
cana-843	119	28	𝐺	𝐺	NOUN
cana-843	119	29	so	so	SCONJ
cana-843	119	30	that	that	SCONJ
cana-843	119	31	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	119	32	)	)	PUNCT
cana-843	119	33	=	=	SYM
cana-843	119	34	1	1	X
cana-843	119	35	.	.	PUNCT
cana-843	119	36	”	"	PUNCT
cana-843	120	1	thus	thus	ADV
cana-843	120	2	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	120	3	)	)	PUNCT
cana-843	120	4	>	>	X
cana-843	120	5	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	120	6	)	)	PUNCT
cana-843	120	7	.	.	PUNCT
cana-843	121	1	theorem	theorem	VERB
cana-843	121	2	3.3	3.3	NUM
cana-843	121	3	.	.	PUNCT
cana-843	122	1	in	in	ADP
cana-843	122	2	a	a	DET
cana-843	122	3	connected	connected	ADJ
cana-843	122	4	graph	graph	NOUN
cana-843	122	5	𝐺	𝐺	NOUN
cana-843	122	6	,	,	PUNCT
cana-843	122	7	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	122	8	)	)	PUNCT
cana-843	122	9	=	=	SYM
cana-843	122	10	𝑎	𝑎	PROPN
cana-843	122	11	and	and	CCONJ
cana-843	122	12	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	122	13	)	)	PUNCT
cana-843	122	14	=	=	SYM
cana-843	122	15	0	0	NUM
cana-843	122	16	exist	exist	VERB
cana-843	122	17	for	for	ADP
cana-843	122	18	each	each	DET
cana-843	122	19	integer	integer	NOUN
cana-843	122	20	𝑎	𝑎	PRON
cana-843	122	21	≥	≥	NOUN
cana-843	122	22	0	0	NUM
cana-843	122	23	.	.	PUNCT
cana-843	123	1	proof	proof	NOUN
cana-843	123	2	.	.	PUNCT
cana-843	124	1	assume	assume	VERB
cana-843	124	2	that	that	SCONJ
cana-843	124	3	𝑃	𝑃	NOUN
cana-843	124	4	:	:	PUNCT
cana-843	124	5	𝑢	𝑢	X
cana-843	124	6	,	,	PUNCT
cana-843	124	7	𝑣	𝑣	NOUN
cana-843	124	8	,	,	PUNCT
cana-843	124	9	𝑤	𝑤	ADP
cana-843	124	10	,	,	PUNCT
cana-843	124	11	𝑥	𝑥	PRON
cana-843	124	12	is	be	AUX
cana-843	124	13	an	an	DET
cana-843	124	14	order	order	NOUN
cana-843	124	15	four	four	NUM
cana-843	124	16	path	path	NOUN
cana-843	124	17	.	.	PUNCT
cana-843	125	1	consider	consider	VERB
cana-843	125	2	𝑃𝑖	𝑃𝑖	VERB
cana-843	125	3	:	:	PUNCT
cana-843	125	4	𝑢𝑖	𝑢𝑖	NOUN
cana-843	125	5	,	,	PUNCT
cana-843	125	6	𝑣𝑖	𝑣𝑖	ADP
cana-843	125	7	(	(	PUNCT
cana-843	125	8	1	1	NUM
cana-843	125	9	≤	≤	NUM
cana-843	125	10	𝑖	𝑖	SYM
cana-843	125	11	≤	≤	NUM
cana-843	125	12	𝑎	𝑎	X
cana-843	125	13	)	)	PUNCT
cana-843	125	14	represent	represent	VERB
cana-843	125	15	an	an	DET
cana-843	125	16	identical	identical	ADJ
cana-843	125	17	pair	pair	NOUN
cana-843	125	18	of	of	ADP
cana-843	125	19	vertices	vertex	NOUN
cana-843	125	20	.	.	PUNCT
cana-843	126	1	let	let	VERB
cana-843	126	2	𝐺	𝐺	PRON
cana-843	126	3	be	be	AUX
cana-843	126	4	the	the	DET
cana-843	126	5	graph	graph	NOUN
cana-843	126	6	generated	generate	VERB
cana-843	126	7	by	by	ADP
cana-843	126	8	adding	add	VERB
cana-843	126	9	the	the	DET
cana-843	126	10	edges	edge	NOUN
cana-843	126	11	𝑣𝑢𝑖	𝑣𝑢𝑖	ADJ
cana-843	126	12	and	and	CCONJ
cana-843	126	13	𝑤𝑣𝑖	𝑤𝑣𝑖	NOUN
cana-843	126	14	to	to	PART
cana-843	126	15	𝑃	𝑃	VERB
cana-843	126	16	and	and	CCONJ
cana-843	126	17	𝑃𝑖	𝑃𝑖	ADP
cana-843	126	18	(	(	PUNCT
cana-843	126	19	1	1	NUM
cana-843	126	20	≤	≤	NUM
cana-843	126	21	𝑖	𝑖	SYM
cana-843	126	22	≤	≤	NUM
cana-843	126	23	𝑎	𝑎	NUM
cana-843	126	24	)	)	PUNCT
cana-843	126	25	.	.	PUNCT
cana-843	127	1	the	the	DET
cana-843	127	2	figure	figure	NOUN
cana-843	127	3	3.3	3.3	NUM
cana-843	127	4	displays	display	VERB
cana-843	127	5	the	the	DET
cana-843	127	6	graph	graph	NOUN
cana-843	127	7	𝐺.	𝐺.	PROPN
cana-843	127	8	𝐺	𝐺	PROPN
cana-843	127	9	figure	figure	NOUN
cana-843	127	10	3.1	3.1	NUM
cana-843	127	11	𝑣1	𝑣1	NOUN
cana-843	127	12	𝑣2	𝑣2	PROPN
cana-843	127	13	𝑣3	𝑣3	PROPN
cana-843	127	14	𝑣4	𝑣4	NOUN
cana-843	127	15	𝑣5	𝑣5	PROPN
cana-843	127	16	𝐺	𝐺	PROPN
cana-843	127	17	figure	figure	NOUN
cana-843	127	18	3.2	3.2	NUM
cana-843	127	19	𝑣1	𝑣1	NOUN
cana-843	127	20	𝑣2	𝑣2	PROPN
cana-843	127	21	𝑣3	𝑣3	PROPN
cana-843	127	22	𝑣4	𝑣4	VERB
cana-843	127	23	𝑣5	𝑣5	NOUN
cana-843	127	24	𝑣7	𝑣7	ADJ
cana-843	127	25	𝑣6	𝑣6	PROPN
cana-843	127	26	communications	communication	NOUN
cana-843	127	27	on	on	ADP
cana-843	127	28	applied	apply	VERB
cana-843	127	29	nonlinear	nonlinear	ADJ
cana-843	127	30	analysis	analysis	NOUN
cana-843	127	31	issn	issn	NOUN
cana-843	127	32	:	:	PUNCT
cana-843	127	33	1074	1074	NUM
cana-843	127	34	-	-	PUNCT
cana-843	127	35	133x	133x	NUM
cana-843	127	36	vol	vol	NOUN
cana-843	127	37	31	31	NUM
cana-843	127	38	no	no	NOUN
cana-843	127	39	.	.	PUNCT
cana-843	128	1	4s	4s	NUM
cana-843	128	2	(	(	PUNCT
cana-843	128	3	2024	2024	NUM
cana-843	128	4	)	)	PUNCT
cana-843	128	5	224	224	NUM
cana-843	128	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	128	7	we	we	PRON
cana-843	128	8	first	first	ADV
cana-843	128	9	establish	establish	VERB
cana-843	128	10	that	that	SCONJ
cana-843	128	11	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	128	12	)	)	PUNCT
cana-843	128	13	=	=	PUNCT
cana-843	128	14	𝑎.	𝑎.	NOUN
cana-843	128	15	let	let	VERB
cana-843	128	16	𝑍	𝑍	VERB
cana-843	128	17	=	=	SYM
cana-843	128	18	{	{	PUNCT
cana-843	128	19	𝑢	𝑢	X
cana-843	128	20	,	,	PUNCT
cana-843	128	21	𝑥	𝑥	PRON
cana-843	128	22	}	}	PUNCT
cana-843	128	23	represent	represent	VERB
cana-843	128	24	all	all	PRON
cana-843	128	25	of	of	ADP
cana-843	128	26	𝐺	𝐺	PROPN
cana-843	128	27	's	's	PART
cana-843	128	28	end	end	NOUN
cana-843	128	29	vertices	vertex	NOUN
cana-843	128	30	.	.	PUNCT
cana-843	129	1	𝑍	𝑍	NOUN
cana-843	129	2	is	be	AUX
cana-843	129	3	a	a	DET
cana-843	129	4	subset	subset	NOUN
cana-843	129	5	of	of	ADP
cana-843	129	6	every	every	DET
cana-843	129	7	𝑔-set	𝑔-set	NOUN
cana-843	129	8	in	in	ADP
cana-843	129	9	𝐺	𝐺	PROPN
cana-843	129	10	,	,	PUNCT
cana-843	129	11	according	accord	VERB
cana-843	129	12	to	to	ADP
cana-843	129	13	theorem	theorem	ADJ
cana-843	129	14	1.1	1.1	NUM
cana-843	129	15	.	.	PUNCT
cana-843	130	1	for	for	ADP
cana-843	130	2	(	(	PUNCT
cana-843	130	3	1	1	NUM
cana-843	130	4	≤	≤	NUM
cana-843	130	5	𝑖	𝑖	SYM
cana-843	130	6	≤	≤	NUM
cana-843	130	7	𝑎	𝑎	X
cana-843	130	8	)	)	PUNCT
cana-843	130	9	,	,	PUNCT
cana-843	130	10	consider	consider	VERB
cana-843	130	11	𝐻𝑖	𝐻𝑖	PROPN
cana-843	130	12	=	=	SYM
cana-843	130	13	{	{	PUNCT
cana-843	130	14	𝑢𝑖	𝑢𝑖	INTJ
cana-843	130	15	,	,	PUNCT
cana-843	130	16	𝑣𝑖	𝑣𝑖	ADP
cana-843	130	17	}	}	PUNCT
cana-843	130	18	.	.	PUNCT
cana-843	131	1	it	it	PRON
cana-843	131	2	is	be	AUX
cana-843	131	3	easily	easily	ADV
cana-843	131	4	shown	show	VERB
cana-843	131	5	that	that	SCONJ
cana-843	131	6	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	131	7	)	)	PUNCT
cana-843	131	8	≥	≥	NOUN
cana-843	131	9	𝑎	𝑎	NOUN
cana-843	131	10	since	since	SCONJ
cana-843	131	11	every	every	DET
cana-843	131	12	vertex	vertex	NOUN
cana-843	131	13	in	in	ADP
cana-843	131	14	the	the	DET
cana-843	131	15	𝑔-set	𝑔-set	NOUN
cana-843	131	16	of	of	ADP
cana-843	131	17	𝐺	𝐺	PROPN
cana-843	131	18	contains	contain	VERB
cana-843	131	19	exactly	exactly	ADV
cana-843	131	20	one	one	NUM
cana-843	131	21	vertex	vertex	NOUN
cana-843	131	22	from	from	ADP
cana-843	131	23	each	each	DET
cana-843	131	24	𝐻𝑖(1	𝐻𝑖(1	ADJ
cana-843	131	25	≤	≤	NUM
cana-843	131	26	𝑖	𝑖	SYM
cana-843	131	27	≤	≤	NUM
cana-843	132	1	𝑎	𝑎	NUM
cana-843	132	2	)	)	PUNCT
cana-843	132	3	.	.	PUNCT
cana-843	133	1	let	let	VERB
cana-843	133	2	𝑆	𝑆	PROPN
cana-843	133	3	=	=	SYM
cana-843	133	4	𝑍	𝑍	PROPN
cana-843	133	5	∪	∪	ADJ
cana-843	133	6	{	{	PUNCT
cana-843	133	7	𝑢1	𝑢1	NOUN
cana-843	133	8	,	,	PUNCT
cana-843	133	9	𝑢2	𝑢2	PROPN
cana-843	133	10	,	,	PUNCT
cana-843	133	11	…	…	PUNCT
cana-843	133	12	,	,	PUNCT
cana-843	133	13	𝑢𝑎	𝑢𝑎	NOUN
cana-843	133	14	}	}	PUNCT
cana-843	133	15	.	.	PUNCT
cana-843	134	1	as	as	ADP
cana-843	134	2	a	a	DET
cana-843	134	3	result	result	NOUN
cana-843	134	4	,	,	PUNCT
cana-843	134	5	𝑆	𝑆	PROPN
cana-843	134	6	is	be	AUX
cana-843	134	7	a	a	DET
cana-843	134	8	𝑔-set	𝑔-set	NOUN
cana-843	134	9	of	of	ADP
cana-843	134	10	𝐺	𝐺	PROPN
cana-843	134	11	and	and	CCONJ
cana-843	134	12	𝑔(𝐺	𝑔(𝐺	PROPN
cana-843	134	13	)	)	PUNCT
cana-843	135	1	=	=	PUNCT
cana-843	136	1	𝑎	𝑎	X
cana-843	136	2	+	+	NUM
cana-843	136	3	2	2	NUM
cana-843	136	4	,	,	PUNCT
cana-843	136	5	as	as	ADP
cana-843	136	6	𝐼[𝑆	𝐼[𝑆	NOUN
cana-843	136	7	]	]	X
cana-843	136	8	=	=	SYM
cana-843	136	9	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	136	10	)	)	PUNCT
cana-843	136	11	.	.	PUNCT
cana-843	137	1	for	for	ADP
cana-843	137	2	every	every	DET
cana-843	137	3	𝑔-set	𝑔-set	NOUN
cana-843	137	4	of	of	ADP
cana-843	137	5	𝐺	𝐺	PROPN
cana-843	137	6	contains	contain	VERB
cana-843	137	7	a	a	DET
cana-843	137	8	subset	subset	NOUN
cana-843	137	9	,	,	PUNCT
cana-843	137	10	𝑍.	𝑍.	VERB
cana-843	137	11	by	by	ADP
cana-843	137	12	theorem	theorem	ADJ
cana-843	137	13	1.2	1.2	NUM
cana-843	137	14	,	,	PUNCT
cana-843	137	15	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	137	16	)	)	PUNCT
cana-843	137	17	≤	≤	NUM
cana-843	137	18	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	137	19	)	)	PUNCT
cana-843	138	1	−	−	NOUN
cana-843	138	2	|𝑍|	|𝑍|	NOUN
cana-843	139	1	=	=	PUNCT
cana-843	140	1	𝑎	𝑎	X
cana-843	140	2	+	+	NUM
cana-843	140	3	2	2	NUM
cana-843	140	4	−	−	NOUN
cana-843	140	5	2	2	NUM
cana-843	140	6	=	=	SYM
cana-843	140	7	𝑎.	𝑎.	VERB
cana-843	140	8	therefore	therefore	ADV
cana-843	140	9	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	140	10	)	)	PUNCT
cana-843	140	11	≤	≤	NOUN
cana-843	140	12	𝑎.	𝑎.	NOUN
cana-843	140	13	considering	consider	VERB
cana-843	140	14	that	that	SCONJ
cana-843	140	15	𝑔(a𝐺	𝑔(a𝐺	PROPN
cana-843	140	16	)	)	PUNCT
cana-843	141	1	=	=	SYM
cana-843	141	2	a𝑎	a𝑎	PROPN
cana-843	141	3	+	+	NOUN
cana-843	141	4	2	2	NUM
cana-843	141	5	and	and	CCONJ
cana-843	141	6	that	that	DET
cana-843	141	7	𝑍	𝑍	NOUN
cana-843	141	8	exists	exist	VERB
cana-843	141	9	in	in	ADP
cana-843	141	10	every	every	DET
cana-843	141	11	𝑔-set	𝑔-set	NOUN
cana-843	141	12	of	of	ADP
cana-843	141	13	𝐺	𝐺	PROPN
cana-843	141	14	,	,	PUNCT
cana-843	141	15	following	follow	VERB
cana-843	141	16	that	that	SCONJ
cana-843	141	17	each	each	DET
cana-843	141	18	𝑔-set	𝑔-set	NOUN
cana-843	141	19	of	of	ADP
cana-843	141	20	𝐺	𝐺	PROPN
cana-843	141	21	,	,	PUNCT
cana-843	141	22	if	if	SCONJ
cana-843	141	23	so	so	ADV
cana-843	141	24	,	,	PUNCT
cana-843	141	25	has	have	VERB
cana-843	141	26	the	the	DET
cana-843	141	27	form	form	NOUN
cana-843	141	28	𝑆	𝑆	PROPN
cana-843	141	29	=	=	SYM
cana-843	141	30	𝑍	𝑍	PROPN
cana-843	141	31	∪	∪	X
cana-843	141	32	{	{	PUNCT
cana-843	141	33	a𝑐1	a𝑐1	NOUN
cana-843	141	34	,	,	PUNCT
cana-843	141	35	a𝑐2	a𝑐2	PROPN
cana-843	141	36	,	,	PUNCT
cana-843	141	37	…	…	PUNCT
cana-843	141	38	,	,	PUNCT
cana-843	141	39	a𝑐𝑎	a𝑐𝑎	PROPN
cana-843	141	40	}	}	PUNCT
cana-843	141	41	,	,	PUNCT
cana-843	141	42	where	where	SCONJ
cana-843	141	43	𝑐𝑖	𝑐𝑖	PROPN
cana-843	141	44	∈	∈	PROPN
cana-843	141	45	𝐻𝑖(1	𝐻𝑖(1	PROPN
cana-843	141	46	≤	≤	NUM
cana-843	141	47	𝑖	𝑖	SYM
cana-843	141	48	≤	≤	NUM
cana-843	141	49	𝑎	𝑎	X
cana-843	141	50	)	)	PUNCT
cana-843	141	51	.	.	PUNCT
cana-843	142	1	given	give	VERB
cana-843	142	2	|𝑇|	|𝑇|	NOUN
cana-843	142	3	<	<	X
cana-843	142	4	𝑎	𝑎	NOUN
cana-843	142	5	,	,	PUNCT
cana-843	142	6	let	let	VERB
cana-843	142	7	𝑇	𝑇	PROPN
cana-843	142	8	be	be	AUX
cana-843	142	9	any	any	DET
cana-843	142	10	proper	proper	ADJ
cana-843	142	11	subset	subset	NOUN
cana-843	142	12	of	of	ADP
cana-843	142	13	𝑆.	𝑆.	PROPN
cana-843	142	14	after	after	ADP
cana-843	142	15	that	that	PRON
cana-843	142	16	,	,	PUNCT
cana-843	142	17	a𝑐𝑗	a𝑐𝑗	PROPN
cana-843	142	18	(	(	PUNCT
cana-843	142	19	1	1	NUM
cana-843	142	20	≤	≤	NOUN
cana-843	142	21	a𝑗	a𝑗	ADP
cana-843	142	22	≤	≤	NUM
cana-843	142	23	a𝑎	a𝑎	PROPN
cana-843	142	24	)	)	PUNCT
cana-843	142	25	is	be	AUX
cana-843	142	26	a	a	DET
cana-843	142	27	vertex	vertex	NOUN
cana-843	142	28	such	such	ADJ
cana-843	142	29	that	that	SCONJ
cana-843	142	30	a𝑐	a𝑐	PROPN
cana-843	142	31	𝑗	𝑗	PROPN
cana-843	142	32	∉	∉	PROPN
cana-843	142	33	a𝑇.	a𝑇.	PROPN
cana-843	142	34	assume	assume	VERB
cana-843	142	35	that	that	SCONJ
cana-843	142	36	𝑏𝑗	𝑏𝑗	PROPN
cana-843	142	37	,	,	PUNCT
cana-843	142	38	a	a	DET
cana-843	142	39	vertex	vertex	NOUN
cana-843	142	40	of	of	ADP
cana-843	142	41	𝐻𝑗	𝐻𝑗	PROPN
cana-843	142	42	,	,	PUNCT
cana-843	142	43	is	be	AUX
cana-843	142	44	distinct	distinct	ADJ
cana-843	142	45	from	from	ADP
cana-843	142	46	a𝑐	a𝑐	PROPN
cana-843	142	47	𝑗	𝑗	INTJ
cana-843	142	48	.	.	PUNCT
cana-843	143	1	consequently	consequently	ADV
cana-843	143	2	,	,	PUNCT
cana-843	143	3	a𝑆	a𝑆	PROPN
cana-843	143	4	1	1	NUM
cana-843	143	5	=	=	SYM
cana-843	143	6	(	(	PUNCT
cana-843	143	7	a𝑆−	a𝑆−	PROPN
cana-843	143	8	{	{	PUNCT
cana-843	143	9	a𝑐	a𝑐	NOUN
cana-843	143	10	𝑗	𝑗	INTJ
cana-843	143	11	}	}	PUNCT
cana-843	143	12	)	)	PUNCT
cana-843	143	13	∪	∪	NOUN
cana-843	143	14	{	{	PUNCT
cana-843	143	15	a𝑏	a𝑏	INTJ
cana-843	143	16	𝑗	𝑗	INTJ
cana-843	143	17	}	}	PUNCT
cana-843	143	18	is	be	AUX
cana-843	143	19	a	a	DET
cana-843	143	20	g	g	NOUN
cana-843	143	21	-	-	PUNCT
cana-843	143	22	set	set	NOUN
cana-843	143	23	that	that	PRON
cana-843	143	24	properly	properly	ADV
cana-843	143	25	contains	contain	VERB
cana-843	143	26	𝑇.	𝑇.	PROPN
cana-843	143	27	as	as	ADP
cana-843	143	28	a	a	DET
cana-843	143	29	result	result	NOUN
cana-843	143	30	,	,	PUNCT
cana-843	143	31	𝑇	𝑇	PROPN
cana-843	143	32	is	be	AUX
cana-843	143	33	not	not	PART
cana-843	143	34	a	a	DET
cana-843	143	35	forcing	force	VERB
cana-843	143	36	subset	subset	NOUN
cana-843	143	37	of	of	ADP
cana-843	143	38	s.	s.	PROPN
cana-843	143	39	for	for	ADP
cana-843	143	40	every	every	DET
cana-843	143	41	minimum	minimum	ADJ
cana-843	143	42	𝑔-set	𝑔-set	NOUN
cana-843	143	43	of	of	ADP
cana-843	143	44	𝐺	𝐺	PROPN
cana-843	143	45	,	,	PUNCT
cana-843	143	46	this	this	PRON
cana-843	143	47	holds	hold	VERB
cana-843	143	48	true	true	ADJ
cana-843	143	49	.	.	PUNCT
cana-843	144	1	therefore	therefore	ADV
cana-843	144	2	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	144	3	)	)	PUNCT
cana-843	144	4	=	=	PUNCT
cana-843	145	1	𝑎.	𝑎.	VERB
cana-843	145	2	next	next	ADJ
cana-843	145	3	we	we	PRON
cana-843	145	4	prove	prove	VERB
cana-843	145	5	that	that	SCONJ
cana-843	145	6	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	145	7	)	)	PUNCT
cana-843	145	8	=	=	SYM
cana-843	146	1	0	0	X
cana-843	146	2	.	.	PUNCT
cana-843	147	1	since	since	SCONJ
cana-843	147	2	𝑍	𝑍	PROPN
cana-843	147	3	is	be	AUX
cana-843	147	4	the	the	DET
cana-843	147	5	distinct	distinct	ADJ
cana-843	147	6	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	147	7	of	of	ADP
cana-843	147	8	𝐺	𝐺	PROPN
cana-843	147	9	,	,	PUNCT
cana-843	147	10	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	147	11	)	)	PUNCT
cana-843	148	1	=	=	SYM
cana-843	148	2	0	0	X
cana-843	148	3	.	.	X
cana-843	148	4	figure	figure	VERB
cana-843	148	5	3.3	3.3	NUM
cana-843	148	6	theorem	theorem	NOUN
cana-843	148	7	3.4	3.4	NUM
cana-843	148	8	.	.	PUNCT
cana-843	149	1	for	for	ADP
cana-843	149	2	every	every	DET
cana-843	149	3	integer	integer	NOUN
cana-843	149	4	𝑎	𝑎	PRON
cana-843	149	5	≥	≥	NOUN
cana-843	149	6	0	0	NUM
cana-843	149	7	,	,	PUNCT
cana-843	149	8	“	"	PUNCT
cana-843	149	9	there	there	PRON
cana-843	149	10	exists	exist	VERB
cana-843	149	11	a	a	DET
cana-843	149	12	connected	connected	ADJ
cana-843	149	13	graph	graph	NOUN
cana-843	149	14	𝐺	𝐺	PROPN
cana-843	149	15	such	such	ADJ
cana-843	149	16	that	that	DET
cana-843	149	17	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	149	18	)	)	PUNCT
cana-843	149	19	=	=	SYM
cana-843	149	20	0	0	NUM
cana-843	149	21	and	and	CCONJ
cana-843	149	22	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	149	23	)	)	PUNCT
cana-843	149	24	=	=	SYM
cana-843	149	25	𝑎.	𝑎.	NOUN
cana-843	149	26	for	for	ADP
cana-843	149	27	every	every	DET
cana-843	149	28	integer	integer	NOUN
cana-843	149	29	𝑎	𝑎	PRON
cana-843	149	30	≥	≥	NOUN
cana-843	149	31	0	0	NUM
cana-843	149	32	,	,	PUNCT
cana-843	149	33	there	there	PRON
cana-843	149	34	exists	exist	VERB
cana-843	149	35	a	a	DET
cana-843	149	36	connected	connected	ADJ
cana-843	149	37	graph	graph	NOUN
cana-843	149	38	𝐺	𝐺	PROPN
cana-843	149	39	such	such	ADJ
cana-843	149	40	that	that	DET
cana-843	149	41	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	149	42	)	)	PUNCT
cana-843	149	43	=	=	SYM
cana-843	149	44	𝑎	𝑎	PROPN
cana-843	149	45	and	and	CCONJ
cana-843	149	46	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	149	47	)	)	PUNCT
cana-843	149	48	=	=	SYM
cana-843	149	49	𝑎.	𝑎.	NOUN
cana-843	149	50	proof	proof	NOUN
cana-843	149	51	.	.	PUNCT
cana-843	150	1	let	let	VERB
cana-843	150	2	𝑃′	𝑃′	NOUN
cana-843	150	3	:	:	PUNCT
cana-843	150	4	𝑤1	𝑤1	VERB
cana-843	150	5	,	,	PUNCT
cana-843	150	6	𝑤2	𝑤2	NOUN
cana-843	150	7	,	,	PUNCT
cana-843	150	8	𝑤3	𝑤3	PROPN
cana-843	150	9	be	be	AUX
cana-843	150	10	a	a	DET
cana-843	150	11	path	path	NOUN
cana-843	150	12	of	of	ADP
cana-843	150	13	order	order	NOUN
cana-843	150	14	3	3	NUM
cana-843	150	15	,	,	PUNCT
cana-843	150	16	and	and	CCONJ
cana-843	150	17	“	"	PUNCT
cana-843	150	18	consider	consider	VERB
cana-843	150	19	𝑃𝑖	𝑃𝑖	ADP
cana-843	150	20	:	:	PUNCT
cana-843	150	21	𝑡1	𝑡1	NOUN
cana-843	150	22	,	,	PUNCT
cana-843	150	23	𝑡2	𝑡2	PROPN
cana-843	150	24	,	,	PUNCT
cana-843	150	25	𝑡3	𝑡3	PROPN
cana-843	150	26	,	,	PUNCT
cana-843	150	27	𝑡4	𝑡4	PROPN
cana-843	150	28	,	,	PUNCT
cana-843	150	29	𝑡5	𝑡5	X
cana-843	150	30	be	be	VERB
cana-843	150	31	a	a	DET
cana-843	150	32	path	path	NOUN
cana-843	150	33	of	of	ADP
cana-843	150	34	order	order	NOUN
cana-843	150	35	5	5	X
cana-843	150	36	.	.	PUNCT
cana-843	151	1	let	let	VERB
cana-843	151	2	𝑃𝑖e	𝑃𝑖e	NOUN
cana-843	151	3	:	:	PUNCT
cana-843	151	4	𝑟𝑖e	𝑟𝑖e	ADJ
cana-843	151	5	,	,	PUNCT
cana-843	151	6	𝑠𝑖e	𝑠𝑖e	ADJ
cana-843	151	7	(	(	PUNCT
cana-843	151	8	1	1	NUM
cana-843	151	9	≤	≤	NUM
cana-843	151	10	𝑖	𝑖	SYM
cana-843	151	11	≤	≤	NUM
cana-843	151	12	𝑎	𝑎	X
cana-843	151	13	)	)	PUNCT
cana-843	151	14	be	be	VERB
cana-843	151	15	an	an	DET
cana-843	151	16	order	order	NOUN
cana-843	151	17	2	2	NUM
cana-843	151	18	replica	replica	NOUN
cana-843	151	19	of	of	ADP
cana-843	151	20	the	the	DET
cana-843	151	21	path	path	NOUN
cana-843	151	22	.	.	PUNCT
cana-843	152	1	let	let	VERB
cana-843	152	2	𝐺e	𝐺e	NOUN
cana-843	152	3	be	be	AUX
cana-843	152	4	the	the	DET
cana-843	152	5	graph	graph	NOUN
cana-843	152	6	created	create	VERB
cana-843	152	7	by	by	ADP
cana-843	152	8	adding	add	VERB
cana-843	152	9	the	the	DET
cana-843	152	10	edges	edge	NOUN
cana-843	152	11	𝑡2𝑤1	𝑡2𝑤1	NOUN
cana-843	152	12	,	,	PUNCT
cana-843	152	13	𝑡2𝑤2	𝑡2𝑤2	PROPN
cana-843	152	14	,	,	PUNCT
cana-843	152	15	𝑡4𝑤2	𝑡4𝑤2	X
cana-843	152	16	,	,	PUNCT
cana-843	152	17	𝑡4𝑤3	𝑡4𝑤3	NOUN
cana-843	152	18	,	,	PUNCT
cana-843	152	19	𝑡2𝑟𝑖	𝑡2𝑟𝑖	PUNCT
cana-843	152	20	(	(	PUNCT
cana-843	152	21	1	1	NUM
cana-843	152	22	≤	≤	NUM
cana-843	152	23	𝑖	𝑖	SYM
cana-843	152	24	≤	≤	NUM
cana-843	152	25	𝑎	𝑎	NOUN
cana-843	152	26	)	)	PUNCT
cana-843	152	27	and	and	CCONJ
cana-843	152	28	𝑡4𝑆𝑖	𝑡4𝑆𝑖	PROPN
cana-843	152	29	(	(	PUNCT
cana-843	152	30	1	1	NUM
cana-843	152	31	≤	≤	NUM
cana-843	152	32	𝑖	𝑖	SYM
cana-843	152	33	≤	≤	NUM
cana-843	152	34	𝑎	𝑎	X
cana-843	152	35	)	)	PUNCT
cana-843	152	36	to	to	ADP
cana-843	152	37	𝑃′	𝑃′	NOUN
cana-843	152	38	and	and	CCONJ
cana-843	152	39	𝑃𝑖	𝑃𝑖	PROPN
cana-843	152	40	(	(	PUNCT
cana-843	152	41	1	1	NUM
cana-843	152	42	≤	≤	NUM
cana-843	152	43	𝑖	𝑖	SYM
cana-843	152	44	≤	≤	NUM
cana-843	152	45	𝑎	𝑎	NUM
cana-843	152	46	)	)	PUNCT
cana-843	152	47	.	.	PUNCT
cana-843	153	1	the	the	DET
cana-843	153	2	figure	figure	NOUN
cana-843	153	3	3.4	3.4	NUM
cana-843	153	4	displays	display	VERB
cana-843	153	5	the	the	DET
cana-843	153	6	graph	graph	NOUN
cana-843	153	7	𝐺.	𝐺.	PROPN
cana-843	153	8	first	first	ADV
cana-843	153	9	,	,	PUNCT
cana-843	153	10	we	we	PRON
cana-843	153	11	establish	establish	VERB
cana-843	153	12	that	that	SCONJ
cana-843	153	13	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	153	14	)	)	PUNCT
cana-843	154	1	=	=	PUNCT
cana-843	154	2	𝑎.	𝑎.	NOUN
cana-843	154	3	let	let	VERB
cana-843	154	4	the	the	DET
cana-843	154	5	set	set	NOUN
cana-843	154	6	of	of	ADP
cana-843	154	7	all	all	PRON
cana-843	154	8	of	of	ADP
cana-843	154	9	𝐺	𝐺	PROPN
cana-843	154	10	's	's	PART
cana-843	154	11	end	end	NOUN
cana-843	154	12	vertices	vertex	NOUN
cana-843	154	13	be	be	VERB
cana-843	154	14	𝑍	𝑍	NOUN
cana-843	154	15	=	=	SYM
cana-843	154	16	{	{	PUNCT
cana-843	154	17	𝑡1	𝑡1	NOUN
cana-843	154	18	,	,	PUNCT
cana-843	154	19	𝑡5	𝑡5	PROPN
cana-843	154	20	}	}	PUNCT
cana-843	154	21	.	.	PUNCT
cana-843	155	1	such	such	ADJ
cana-843	155	2	that	that	DET
cana-843	155	3	𝑍	𝑍	PROPN
cana-843	155	4	is	be	AUX
cana-843	155	5	therefore	therefore	ADV
cana-843	155	6	a	a	DET
cana-843	155	7	subset	subset	NOUN
cana-843	155	8	of	of	ADP
cana-843	155	9	each	each	DET
cana-843	155	10	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	155	11	of	of	ADP
cana-843	155	12	𝐺	𝐺	PROPN
cana-843	155	13	according	accord	VERB
cana-843	155	14	to	to	ADP
cana-843	155	15	theorem	theorem	ADJ
cana-843	155	16	1.1	1.1	NUM
cana-843	155	17	.	.	PUNCT
cana-843	156	1	𝐻𝑖e	𝐻𝑖e	PROPN
cana-843	156	2	:	:	PUNCT
cana-843	156	3	{	{	PUNCT
cana-843	156	4	𝑟𝑖	𝑟𝑖	X
cana-843	156	5	e	e	NOUN
cana-843	156	6	,	,	PUNCT
cana-843	156	7	𝑠𝑖e	𝑠𝑖e	ADJ
cana-843	156	8	}	}	PUNCT
cana-843	156	9	(	(	PUNCT
cana-843	156	10	1	1	NUM
cana-843	156	11	≤	≤	NUM
cana-843	156	12	𝑖	𝑖	SYM
cana-843	156	13	≤	≤	NUM
cana-843	156	14	𝑎	𝑎	X
cana-843	156	15	)	)	PUNCT
cana-843	156	16	be	be	AUX
cana-843	156	17	given	give	VERB
cana-843	156	18	.	.	PUNCT
cana-843	157	1	then	then	ADV
cana-843	157	2	,	,	PUNCT
cana-843	157	3	it	it	PRON
cana-843	157	4	is	be	AUX
cana-843	157	5	evident	evident	ADJ
cana-843	157	6	that	that	SCONJ
cana-843	157	7	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	157	8	)	)	PUNCT
cana-843	157	9	≥	≥	NOUN
cana-843	157	10	𝑎	𝑎	NOUN
cana-843	157	11	+	+	NOUN
cana-843	157	12	2	2	NUM
cana-843	157	13	since	since	SCONJ
cana-843	157	14	every	every	DET
cana-843	157	15	circular	circular	ADJ
cana-843	157	16	set	set	NOUN
cana-843	157	17	of	of	ADP
cana-843	157	18	𝐺	𝐺	PROPN
cana-843	157	19	has	have	AUX
cana-843	157	20	at	at	ADV
cana-843	157	21	least	least	ADV
cana-843	157	22	one	one	NUM
cana-843	157	23	vertex	vertex	NOUN
cana-843	157	24	from	from	ADP
cana-843	157	25	communications	communication	NOUN
cana-843	157	26	on	on	ADP
cana-843	157	27	applied	apply	VERB
cana-843	157	28	nonlinear	nonlinear	ADJ
cana-843	157	29	analysis	analysis	NOUN
cana-843	157	30	issn	issn	NOUN
cana-843	157	31	:	:	PUNCT
cana-843	157	32	1074	1074	NUM
cana-843	157	33	-	-	PUNCT
cana-843	157	34	133x	133x	NUM
cana-843	157	35	vol	vol	NOUN
cana-843	157	36	31	31	NUM
cana-843	157	37	no	no	NOUN
cana-843	157	38	.	.	PUNCT
cana-843	158	1	4s	4s	NUM
cana-843	158	2	(	(	PUNCT
cana-843	158	3	2024	2024	NUM
cana-843	158	4	)	)	PUNCT
cana-843	158	5	225	225	NUM
cana-843	158	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	158	7	each	each	DET
cana-843	158	8	𝐻𝑖	𝐻𝑖	PROPN
cana-843	158	9	(	(	PUNCT
cana-843	158	10	1	1	NUM
cana-843	158	11	≤	≤	NUM
cana-843	158	12	𝑖	𝑖	SYM
cana-843	158	13	≤	≤	NUM
cana-843	158	14	𝑎	𝑎	X
cana-843	158	15	)	)	PUNCT
cana-843	158	16	.	.	PUNCT
cana-843	159	1	let	let	VERB
cana-843	159	2	𝑆	𝑆	PROPN
cana-843	159	3	=	=	SYM
cana-843	159	4	𝑍	𝑍	PROPN
cana-843	159	5	∪	∪	X
cana-843	159	6	{	{	PUNCT
cana-843	159	7	𝑟1	𝑟1	NOUN
cana-843	159	8	,	,	PUNCT
cana-843	159	9	𝑟2	𝑟2	NOUN
cana-843	159	10	,	,	PUNCT
cana-843	159	11	…	…	PUNCT
cana-843	159	12	,	,	PUNCT
cana-843	159	13	𝑟𝑎	𝑟𝑎	NOUN
cana-843	159	14	}	}	PUNCT
cana-843	159	15	.	.	PUNCT
cana-843	160	1	𝐼𝐷𝑐[𝑆	𝐼𝐷𝑐[𝑆	X
cana-843	160	2	]	]	X
cana-843	160	3	=	=	SYM
cana-843	160	4	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	160	5	)	)	PUNCT
cana-843	160	6	in	in	ADP
cana-843	160	7	this	this	DET
cana-843	160	8	case	case	NOUN
cana-843	160	9	,	,	PUNCT
cana-843	160	10	indicating	indicate	VERB
cana-843	160	11	that	that	SCONJ
cana-843	160	12	𝑆	𝑆	PROPN
cana-843	160	13	is	be	AUX
cana-843	160	14	a	a	DET
cana-843	160	15	circular	circular	ADJ
cana-843	160	16	set	set	NOUN
cana-843	160	17	of	of	ADP
cana-843	160	18	𝐺	𝐺	PROPN
cana-843	160	19	and	and	CCONJ
cana-843	160	20	hence	hence	ADV
cana-843	160	21	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NUM
cana-843	160	22	)	)	PUNCT
cana-843	160	23	=	=	PUNCT
cana-843	161	1	𝑎	𝑎	X
cana-843	161	2	+	+	NOUN
cana-843	161	3	2	2	NUM
cana-843	161	4	.	.	NOUN
cana-843	161	5	as	as	SCONJ
cana-843	161	6	𝑍	𝑍	PROPN
cana-843	161	7	is	be	AUX
cana-843	161	8	a	a	DET
cana-843	161	9	subset	subset	NOUN
cana-843	161	10	of	of	ADP
cana-843	161	11	each	each	DET
cana-843	161	12	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	161	13	of	of	ADP
cana-843	161	14	𝐺e	𝐺e	PROPN
cana-843	161	15	,	,	PUNCT
cana-843	161	16	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	161	17	)	)	PUNCT
cana-843	161	18	≤	≤	NOUN
cana-843	161	19	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	161	20	)	)	PUNCT
cana-843	161	21	−	−	NOUN
cana-843	161	22	|𝑍|	|𝑍|	NOUN
cana-843	162	1	=	=	PUNCT
cana-843	163	1	𝑎	𝑎	X
cana-843	163	2	+	+	NUM
cana-843	163	3	2	2	NUM
cana-843	163	4	−	−	NOUN
cana-843	163	5	2	2	NUM
cana-843	163	6	=	=	SYM
cana-843	163	7	𝑎	𝑎	NOUN
cana-843	163	8	,	,	PUNCT
cana-843	163	9	according	accord	VERB
cana-843	163	10	to	to	ADP
cana-843	163	11	theorem	theorem	ADJ
cana-843	163	12	2.3	2.3	NUM
cana-843	163	13	.	.	PUNCT
cana-843	164	1	consequently	consequently	ADV
cana-843	164	2	,	,	PUNCT
cana-843	164	3	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	164	4	)	)	PUNCT
cana-843	164	5	≤	≤	NOUN
cana-843	164	6	𝑎.	𝑎.	NOUN
cana-843	164	7	given	give	VERB
cana-843	164	8	that	that	PRON
cana-843	164	9	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	164	10	)	)	PUNCT
cana-843	164	11	=	=	PUNCT
cana-843	165	1	𝑎	𝑎	NOUN
cana-843	165	2	=	=	SYM
cana-843	165	3	2	2	NUM
cana-843	165	4	,	,	PUNCT
cana-843	165	5	furthermore	furthermore	ADV
cana-843	165	6	,	,	PUNCT
cana-843	165	7	it	it	PRON
cana-843	165	8	is	be	AUX
cana-843	165	9	evident	evident	ADJ
cana-843	165	10	that	that	SCONJ
cana-843	165	11	every	every	DET
cana-843	165	12	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	165	13	of	of	ADP
cana-843	165	14	𝐺	𝐺	PROPN
cana-843	165	15	that	that	PRON
cana-843	165	16	contains	contain	VERB
cana-843	165	17	𝑍	𝑍	PROPN
cana-843	165	18	has	have	VERB
cana-843	165	19	the	the	DET
cana-843	165	20	form	form	NOUN
cana-843	165	21	𝑆	𝑆	PROPN
cana-843	165	22	=	=	PUNCT
cana-843	166	1	𝑍e	𝑍e	VERB
cana-843	166	2	∪	∪	ADV
cana-843	166	3	{	{	PUNCT
cana-843	166	4	𝑐1	𝑐1	NOUN
cana-843	166	5	e	e	NOUN
cana-843	166	6	,	,	PUNCT
cana-843	166	7	𝑐2	𝑐2	NOUN
cana-843	166	8	,	,	PUNCT
cana-843	166	9	…	…	PUNCT
cana-843	166	10	,	,	PUNCT
cana-843	166	11	𝑐𝑎	𝑐𝑎	NOUN
cana-843	166	12	}	}	PUNCT
cana-843	166	13	,	,	PUNCT
cana-843	166	14	where	where	SCONJ
cana-843	166	15	𝑐𝑖	𝑐𝑖	NOUN
cana-843	166	16	∈	∈	PROPN
cana-843	166	17	𝐻𝑖	𝐻𝑖	PROPN
cana-843	166	18	(	(	PUNCT
cana-843	166	19	1	1	NUM
cana-843	166	20	≤	≤	NUM
cana-843	166	21	𝑖e	𝑖e	NOUN
cana-843	166	22	≤	≤	NUM
cana-843	166	23	𝑎	𝑎	NOUN
cana-843	166	24	)	)	PUNCT
cana-843	166	25	.	.	PUNCT
cana-843	167	1	given	give	VERB
cana-843	167	2	|𝑇|	|𝑇|	PROPN
cana-843	167	3	<	<	X
cana-843	167	4	𝑎e	𝑎e	PROPN
cana-843	167	5	,	,	PUNCT
cana-843	167	6	let	let	VERB
cana-843	167	7	𝑇	𝑇	PROPN
cana-843	167	8	be	be	AUX
cana-843	167	9	any	any	DET
cana-843	167	10	suitable	suitable	ADJ
cana-843	167	11	subset	subset	NOUN
cana-843	167	12	of	of	ADP
cana-843	167	13	𝑆.	𝑆.	PROPN
cana-843	167	14	after	after	ADP
cana-843	167	15	that	that	PRON
cana-843	167	16	,	,	PUNCT
cana-843	167	17	𝑐𝑗	𝑐𝑗	CCONJ
cana-843	167	18	(	(	PUNCT
cana-843	167	19	1	1	NUM
cana-843	167	20	≤	≤	NUM
cana-843	167	21	𝑗e	𝑗e	ADP
cana-843	167	22	≤	≤	NUM
cana-843	167	23	𝑎	𝑎	X
cana-843	167	24	)	)	PUNCT
cana-843	167	25	is	be	AUX
cana-843	167	26	a	a	DET
cana-843	167	27	vertex	vertex	NOUN
cana-843	167	28	such	such	ADJ
cana-843	167	29	that	that	SCONJ
cana-843	167	30	𝑐𝑗	𝑐𝑗	PROPN
cana-843	167	31	∉	∉	PROPN
cana-843	167	32	𝑇.	𝑇.	PROPN
cana-843	167	33	assume	assume	VERB
cana-843	167	34	that	that	SCONJ
cana-843	167	35	𝑏𝑗	𝑏𝑗	PROPN
cana-843	167	36	,	,	PUNCT
cana-843	167	37	a	a	DET
cana-843	167	38	vertex	vertex	NOUN
cana-843	167	39	of	of	ADP
cana-843	167	40	𝐻𝑗	𝐻𝑗	PROPN
cana-843	167	41	,	,	PUNCT
cana-843	167	42	is	be	AUX
cana-843	167	43	different	different	ADJ
cana-843	167	44	from	from	ADP
cana-843	167	45	𝑐𝑗.	𝑐𝑗.	NOUN
cana-843	167	46	subseequently	subseequently	ADV
cana-843	167	47	,	,	PUNCT
cana-843	167	48	𝑆1	𝑆1	NOUN
cana-843	167	49	=	=	SYM
cana-843	167	50	(	(	PUNCT
cana-843	167	51	𝑆	𝑆	PROPN
cana-843	167	52	−	−	PROPN
cana-843	167	53	{	{	PUNCT
cana-843	167	54	𝑐𝑗	𝑐𝑗	NOUN
cana-843	167	55	}	}	PUNCT
cana-843	167	56	)	)	PUNCT
cana-843	167	57	∪	∪	SCONJ
cana-843	167	58	{	{	PUNCT
cana-843	167	59	𝑏𝑗	𝑏𝑗	NOUN
cana-843	167	60	}	}	PUNCT
cana-843	167	61	is	be	AUX
cana-843	167	62	a	a	DET
cana-843	167	63	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	167	64	that	that	PRON
cana-843	167	65	correctly	correctly	ADV
cana-843	167	66	contains	contain	VERB
cana-843	167	67	𝑇.	𝑇.	PROPN
cana-843	167	68	𝑇	𝑇	PROPN
cana-843	167	69	is	be	AUX
cana-843	167	70	not	not	PART
cana-843	167	71	a	a	DET
cana-843	167	72	forced	force	VERB
cana-843	167	73	subset	subset	NOUN
cana-843	167	74	of	of	ADP
cana-843	167	75	𝑆	𝑆	PROPN
cana-843	167	76	”	"	PUNCT
cana-843	167	77	as	as	ADP
cana-843	167	78	a	a	DET
cana-843	167	79	result	result	NOUN
cana-843	167	80	.	.	PUNCT
cana-843	168	1	for	for	ADP
cana-843	168	2	every	every	DET
cana-843	168	3	minimum	minimum	ADJ
cana-843	168	4	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	168	5	of	of	ADP
cana-843	168	6	𝐺	𝐺	PROPN
cana-843	168	7	,	,	PUNCT
cana-843	168	8	this	this	PRON
cana-843	168	9	is	be	AUX
cana-843	168	10	true	true	ADJ
cana-843	168	11	.	.	PUNCT
cana-843	169	1	consequently	consequently	ADV
cana-843	169	2	,	,	PUNCT
cana-843	169	3	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	169	4	)	)	PUNCT
cana-843	169	5	=	=	VERB
cana-843	169	6	𝑎.	𝑎.	VERB
cana-843	169	7	next	next	ADV
cana-843	169	8	,	,	PUNCT
cana-843	169	9	we	we	PRON
cana-843	169	10	prove	prove	VERB
cana-843	169	11	that	that	SCONJ
cana-843	169	12	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	169	13	)	)	PUNCT
cana-843	169	14	=	=	PUNCT
cana-843	169	15	𝑎.	𝑎.	VERB
cana-843	169	16	the	the	DET
cana-843	169	17	representation	representation	NOUN
cana-843	169	18	of	of	ADP
cana-843	169	19	every	every	DET
cana-843	169	20	extreme	extreme	ADJ
cana-843	169	21	vertex	vertex	NOUN
cana-843	169	22	in	in	ADP
cana-843	169	23	𝐺	𝐺	PROPN
cana-843	169	24	is	be	AUX
cana-843	169	25	𝑍1	𝑍1	NOUN
cana-843	169	26	=	=	SYM
cana-843	169	27	𝑍	𝑍	VERB
cana-843	169	28	∪	∪	X
cana-843	169	29	{	{	PUNCT
cana-843	169	30	𝑤1	𝑤1	ADJ
cana-843	169	31	,	,	PUNCT
cana-843	169	32	𝑤3	𝑤3	PROPN
cana-843	169	33	}	}	PUNCT
cana-843	169	34	.	.	PUNCT
cana-843	170	1	theorem	theorem	VERB
cana-843	170	2	1.1	1.1	NUM
cana-843	170	3	states	state	NOUN
cana-843	170	4	that	that	SCONJ
cana-843	170	5	every	every	DET
cana-843	170	6	𝑔-set	𝑔-set	NOUN
cana-843	170	7	in	in	ADP
cana-843	170	8	𝐺	𝐺	PROPN
cana-843	170	9	is	be	AUX
cana-843	170	10	a	a	DET
cana-843	170	11	subset	subset	NOUN
cana-843	170	12	of	of	ADP
cana-843	170	13	𝑍1	𝑍1	PROPN
cana-843	170	14	.	.	PUNCT
cana-843	171	1	give	give	VERB
cana-843	171	2	𝐻𝑖	𝐻𝑖	PROPN
cana-843	171	3	:	:	PUNCT
cana-843	171	4	{	{	PUNCT
cana-843	171	5	𝑟𝑖	𝑟𝑖	PART
cana-843	171	6	e	e	NOUN
cana-843	171	7	,	,	PUNCT
cana-843	171	8	𝑠𝑖	𝑠𝑖	NOUN
cana-843	171	9	e	e	NOUN
cana-843	171	10	}	}	PUNCT
cana-843	171	11	(	(	PUNCT
cana-843	171	12	1	1	NUM
cana-843	171	13	≤	≤	NUM
cana-843	171	14	𝑖e	𝑖e	NOUN
cana-843	171	15	≤	≤	NUM
cana-843	171	16	𝑎	𝑎	NOUN
cana-843	171	17	)	)	PUNCT
cana-843	171	18	.	.	PUNCT
cana-843	172	1	since	since	SCONJ
cana-843	172	2	every	every	DET
cana-843	172	3	𝑔-set	𝑔-set	NOUN
cana-843	172	4	of	of	ADP
cana-843	172	5	𝐺	𝐺	PROPN
cana-843	172	6	contains	contain	VERB
cana-843	172	7	at	at	ADV
cana-843	172	8	least	least	ADV
cana-843	172	9	one	one	NUM
cana-843	172	10	vertex	vertex	NOUN
cana-843	172	11	from	from	ADP
cana-843	172	12	every	every	DET
cana-843	172	13	𝐻𝑖(1	𝐻𝑖(1	PROPN
cana-843	172	14	≤	≤	NUM
cana-843	172	15	𝑖e	𝑖e	VERB
cana-843	172	16	≤	≤	NUM
cana-843	172	17	𝑎	𝑎	NOUN
cana-843	172	18	)	)	PUNCT
cana-843	172	19	,	,	PUNCT
cana-843	172	20	it	it	PRON
cana-843	172	21	is	be	AUX
cana-843	172	22	easy	easy	ADJ
cana-843	172	23	to	to	PART
cana-843	172	24	demonstrate	demonstrate	VERB
cana-843	172	25	that	that	SCONJ
cana-843	172	26	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	172	27	)	)	PUNCT
cana-843	172	28	≥	≥	NOUN
cana-843	173	1	𝑎	𝑎	X
cana-843	173	2	+	+	NOUN
cana-843	173	3	4	4	NUM
cana-843	173	4	.	.	X
cana-843	174	1	𝑆	𝑆	PROPN
cana-843	174	2	=	=	SYM
cana-843	174	3	𝑍1	𝑍1	PROPN
cana-843	174	4	∪	∪	X
cana-843	174	5	{	{	PUNCT
cana-843	174	6	𝑟1	𝑟1	NOUN
cana-843	174	7	,	,	PUNCT
cana-843	174	8	𝑟2	𝑟2	NOUN
cana-843	174	9	,	,	PUNCT
cana-843	174	10	…	…	PUNCT
cana-843	174	11	,	,	PUNCT
cana-843	174	12	𝑟𝑎	𝑟𝑎	NOUN
cana-843	174	13	}	}	PUNCT
cana-843	174	14	is	be	AUX
cana-843	174	15	assumed	assume	VERB
cana-843	174	16	.	.	PUNCT
cana-843	175	1	consequently	consequently	ADV
cana-843	175	2	,	,	PUNCT
cana-843	175	3	since	since	SCONJ
cana-843	175	4	𝐼[𝑆	𝐼[𝑆	NOUN
cana-843	175	5	]	]	X
cana-843	175	6	=	=	SYM
cana-843	175	7	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	175	8	)	)	PUNCT
cana-843	175	9	,	,	PUNCT
cana-843	175	10	𝑆	𝑆	PROPN
cana-843	175	11	is	be	AUX
cana-843	175	12	a	a	DET
cana-843	175	13	𝑔-set	𝑔-set	NOUN
cana-843	175	14	of	of	ADP
cana-843	175	15	𝐺	𝐺	PROPN
cana-843	175	16	and	and	CCONJ
cana-843	175	17	𝑔(𝐺	𝑔(𝐺	PROPN
cana-843	175	18	)	)	PUNCT
cana-843	175	19	=	=	PUNCT
cana-843	176	1	𝑎	𝑎	X
cana-843	176	2	+	+	NOUN
cana-843	176	3	4	4	NUM
cana-843	176	4	.	.	X
cana-843	177	1	all	all	PRON
cana-843	177	2	of	of	ADP
cana-843	177	3	the	the	DET
cana-843	177	4	𝑔-sets	𝑔-set	NOUN
cana-843	177	5	in	in	ADP
cana-843	177	6	𝐺	𝐺	PROPN
cana-843	177	7	have	have	VERB
cana-843	177	8	a	a	DET
cana-843	177	9	subset	subset	NOUN
cana-843	177	10	called	call	VERB
cana-843	177	11	𝑍1	𝑍1	PROPN
cana-843	177	12	.	.	PUNCT
cana-843	178	1	the	the	DET
cana-843	178	2	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	178	3	)	)	PUNCT
cana-843	178	4	≤	≤	NOUN
cana-843	178	5	𝑎	𝑎	DET
cana-843	178	6	theorem	theorem	NOUN
cana-843	178	7	applies	applie	NOUN
cana-843	178	8	.	.	PUNCT
cana-843	179	1	𝑍1	𝑍1	PROPN
cana-843	179	2	appears	appear	VERB
cana-843	179	3	in	in	ADP
cana-843	179	4	every	every	DET
cana-843	179	5	𝑔-set	𝑔-set	NOUN
cana-843	179	6	of	of	ADP
cana-843	179	7	𝐺	𝐺	PROPN
cana-843	179	8	,	,	PUNCT
cana-843	179	9	and	and	CCONJ
cana-843	179	10	since	since	SCONJ
cana-843	179	11	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	179	12	)	)	PUNCT
cana-843	179	13	=	=	PUNCT
cana-843	180	1	𝑎	𝑎	X
cana-843	180	2	+	+	NUM
cana-843	180	3	4	4	NUM
cana-843	180	4	,	,	PUNCT
cana-843	180	5	it	it	PRON
cana-843	180	6	follows	follow	VERB
cana-843	180	7	that	that	SCONJ
cana-843	180	8	every	every	DET
cana-843	180	9	𝑔-set	𝑔-set	NOUN
cana-843	180	10	of	of	ADP
cana-843	180	11	𝐺	𝐺	PROPN
cana-843	180	12	has	have	VERB
cana-843	180	13	the	the	DET
cana-843	180	14	form	form	NOUN
cana-843	180	15	𝑆	𝑆	PROPN
cana-843	180	16	=	=	PUNCT
cana-843	181	1	𝑍1	𝑍1	PROPN
cana-843	181	2	∪	∪	X
cana-843	181	3	{	{	PUNCT
cana-843	181	4	𝑐1	𝑐1	NOUN
cana-843	181	5	,	,	PUNCT
cana-843	181	6	𝑐2	𝑐2	NOUN
cana-843	181	7	e	e	NOUN
cana-843	181	8	,	,	PUNCT
cana-843	181	9	…	…	PUNCT
cana-843	181	10	,	,	PUNCT
cana-843	181	11	𝑐𝑎	𝑐𝑎	PRON
cana-843	181	12	e	e	NOUN
cana-843	181	13	}	}	PUNCT
cana-843	181	14	,	,	PUNCT
cana-843	181	15	where	where	SCONJ
cana-843	181	16	𝑐𝑖	𝑐𝑖	PROPN
cana-843	181	17	∈	∈	PROPN
cana-843	181	18	𝐻𝑖(1	𝐻𝑖(1	PROPN
cana-843	181	19	≤	≤	NUM
cana-843	181	20	𝑖e	𝑖e	X
cana-843	181	21	≤	≤	NOUN
cana-843	181	22	𝑎e	𝑎e	ADP
cana-843	181	23	)	)	PUNCT
cana-843	181	24	.	.	PUNCT
cana-843	182	1	let	let	VERB
cana-843	182	2	𝑇	𝑇	PROPN
cana-843	182	3	be	be	AUX
cana-843	182	4	any	any	DET
cana-843	182	5	appropriate	appropriate	ADJ
cana-843	182	6	subset	subset	NOUN
cana-843	182	7	of	of	ADP
cana-843	182	8	𝑆	𝑆	PROPN
cana-843	182	9	such	such	ADJ
cana-843	182	10	that	that	DET
cana-843	182	11	|𝑇|	|𝑇|	NOUN
cana-843	182	12	<	<	X
cana-843	182	13	𝑎.	𝑎.	NOUN
cana-843	182	14	after	after	ADP
cana-843	182	15	that	that	PRON
cana-843	182	16	,	,	PUNCT
cana-843	182	17	a	a	DET
cana-843	182	18	vertex	vertex	NOUN
cana-843	182	19	such	such	ADJ
cana-843	182	20	that	that	PRON
cana-843	182	21	is	be	AUX
cana-843	182	22	𝑐𝑗	𝑐𝑗	PROPN
cana-843	182	23	∉	∉	PROPN
cana-843	182	24	𝑇	𝑇	PROPN
cana-843	182	25	is	be	AUX
cana-843	182	26	𝑐𝑗	𝑐𝑗	INTJ
cana-843	182	27	(	(	PUNCT
cana-843	182	28	1	1	NUM
cana-843	182	29	≤	≤	NUM
cana-843	182	30	𝑗	𝑗	PRON
cana-843	182	31	≤	≤	NUM
cana-843	182	32	𝑎	𝑎	NOUN
cana-843	182	33	)	)	PUNCT
cana-843	182	34	.	.	PUNCT
cana-843	183	1	presume	presume	VERB
cana-843	183	2	that	that	SCONJ
cana-843	183	3	𝑡𝑗	𝑡𝑗	NOUN
cana-843	183	4	,	,	PUNCT
cana-843	183	5	one	one	NUM
cana-843	183	6	of	of	ADP
cana-843	183	7	𝐻𝑗	𝐻𝑗	PROPN
cana-843	183	8	's	's	PART
cana-843	183	9	vertices	vertex	NOUN
cana-843	183	10	,	,	PUNCT
cana-843	183	11	is	be	AUX
cana-843	183	12	distinct	distinct	ADJ
cana-843	183	13	from	from	ADP
cana-843	183	14	𝑐𝑗.	𝑐𝑗.	NOUN
cana-843	183	15	consequently	consequently	ADV
cana-843	183	16	,	,	PUNCT
cana-843	183	17	a	a	DET
cana-843	183	18	𝑔-set	𝑔-set	NOUN
cana-843	183	19	that	that	PRON
cana-843	183	20	suitably	suitably	ADV
cana-843	183	21	contains	contain	VERB
cana-843	183	22	𝑇	𝑇	PROPN
cana-843	183	23	is	be	AUX
cana-843	183	24	𝑆1	𝑆1	NOUN
cana-843	183	25	=	=	SYM
cana-843	183	26	(	(	PUNCT
cana-843	183	27	𝑆	𝑆	PROPN
cana-843	183	28	−	−	PROPN
cana-843	183	29	{	{	PUNCT
cana-843	183	30	𝑐𝑗	𝑐𝑗	NOUN
cana-843	183	31	}	}	PUNCT
cana-843	183	32	)	)	PUNCT
cana-843	183	33	∪	∪	ADP
cana-843	183	34	{	{	PUNCT
cana-843	183	35	𝑏𝑗	𝑏𝑗	NOUN
cana-843	183	36	}	}	PUNCT
cana-843	183	37	.	.	PUNCT
cana-843	184	1	as	as	ADP
cana-843	184	2	such	such	ADJ
cana-843	184	3	,	,	PUNCT
cana-843	184	4	𝑇	𝑇	PROPN
cana-843	184	5	is	be	AUX
cana-843	184	6	not	not	PART
cana-843	184	7	a	a	DET
cana-843	184	8	forced	force	VERB
cana-843	184	9	subset	subset	NOUN
cana-843	184	10	of	of	ADP
cana-843	184	11	𝑆.	𝑆.	PROPN
cana-843	184	12	this	this	PRON
cana-843	184	13	is	be	AUX
cana-843	184	14	valid	valid	ADJ
cana-843	184	15	for	for	ADP
cana-843	184	16	any	any	DET
cana-843	184	17	smallest	small	ADJ
cana-843	184	18	𝑔-set	𝑔-set	NOUN
cana-843	184	19	of	of	ADP
cana-843	184	20	𝐺.	𝐺.	NOUN
cana-843	184	21	as	as	ADP
cana-843	184	22	a	a	DET
cana-843	184	23	result	result	NOUN
cana-843	184	24	,	,	PUNCT
cana-843	184	25	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	184	26	)	)	PUNCT
cana-843	184	27	=	=	PUNCT
cana-843	184	28	𝑎.	𝑎.	NOUN
cana-843	184	29	figure	figure	VERB
cana-843	184	30	3.4	3.4	NUM
cana-843	184	31	theorem	theorem	NOUN
cana-843	184	32	3.5	3.5	NUM
cana-843	184	33	.	.	PUNCT
cana-843	185	1	let	let	VERB
cana-843	185	2	g	g	PRON
cana-843	185	3	be	be	AUX
cana-843	185	4	a	a	DET
cana-843	185	5	connected	connected	ADJ
cana-843	185	6	graph	graph	NOUN
cana-843	185	7	.	.	PUNCT
cana-843	186	1	for	for	ADP
cana-843	186	2	every	every	DET
cana-843	186	3	integer	integer	NOUN
cana-843	186	4	𝑎	𝑎	PRON
cana-843	186	5	≥	≥	NOUN
cana-843	186	6	0	0	NUM
cana-843	186	7	,	,	PUNCT
cana-843	186	8	and	and	CCONJ
cana-843	186	9	𝑏	𝑏	PRON
cana-843	186	10	≥	≥	NOUN
cana-843	186	11	0	0	NUM
cana-843	186	12	,	,	PUNCT
cana-843	186	13	there	there	PRON
cana-843	186	14	exists	exist	VERB
cana-843	186	15	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	186	16	)	)	PUNCT
cana-843	186	17	=	=	SYM
cana-843	187	1	𝑎	𝑎	NOUN
cana-843	187	2	and	and	CCONJ
cana-843	187	3	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	NUM
cana-843	187	4	)	)	PUNCT
cana-843	188	1	=	=	NOUN
cana-843	188	2	𝑏.	𝑏.	NOUN
cana-843	188	3	proof	proof	NOUN
cana-843	188	4	.	.	PUNCT
cana-843	189	1	case	case	NOUN
cana-843	189	2	(	(	PUNCT
cana-843	189	3	i	i	NOUN
cana-843	189	4	)	)	PUNCT
cana-843	189	5	𝑎	𝑎	X
cana-843	189	6	=	=	SYM
cana-843	189	7	0	0	NUM
cana-843	189	8	,	,	PUNCT
cana-843	189	9	𝑏	𝑏	PRON
cana-843	189	10	≥	≥	NOUN
cana-843	189	11	1	1	NUM
cana-843	189	12	.	.	PUNCT
cana-843	190	1	the	the	DET
cana-843	190	2	graph	graph	NOUN
cana-843	190	3	produced	produce	VERB
cana-843	190	4	in	in	ADP
cana-843	190	5	theorem	theorem	ADJ
cana-843	190	6	3.3	3.3	NUM
cana-843	190	7	meets	meet	VERB
cana-843	190	8	the	the	DET
cana-843	190	9	requisite	requisite	ADJ
cana-843	190	10	requirement	requirement	NOUN
cana-843	190	11	.	.	PUNCT
cana-843	191	1	case	case	NOUN
cana-843	191	2	(	(	PUNCT
cana-843	191	3	ii	ii	NOUN
cana-843	191	4	)	)	PUNCT
cana-843	192	1	𝑎	𝑎	DET
cana-843	192	2	≥	≥	NOUN
cana-843	192	3	1	1	NUM
cana-843	192	4	,	,	PUNCT
cana-843	192	5	𝑏	𝑏	PROPN
cana-843	192	6	=	=	SYM
cana-843	192	7	0	0	NUM
cana-843	192	8	.	.	PUNCT
cana-843	193	1	the	the	DET
cana-843	193	2	graph	graph	NOUN
cana-843	193	3	constructed	construct	VERB
cana-843	193	4	theorem	theorem	VERB
cana-843	193	5	3.3	3.3	NUM
cana-843	193	6	,	,	PUNCT
cana-843	193	7	satisfies	satisfy	VERB
cana-843	193	8	the	the	DET
cana-843	193	9	required	required	ADJ
cana-843	193	10	condition	condition	NOUN
cana-843	193	11	.	.	PUNCT
cana-843	194	1	case	case	NOUN
cana-843	194	2	(	(	PUNCT
cana-843	194	3	iii	iii	NOUN
cana-843	194	4	)	)	PUNCT
cana-843	194	5	𝑎	𝑎	NOUN
cana-843	194	6	=	=	SYM
cana-843	194	7	𝑏	𝑏	NOUN
cana-843	194	8	≥	≥	NOUN
cana-843	194	9	1	1	NUM
cana-843	194	10	.	.	PUNCT
cana-843	195	1	the	the	DET
cana-843	195	2	graph	graph	NOUN
cana-843	195	3	constructed	construct	VERB
cana-843	195	4	theorem	theorem	VERB
cana-843	195	5	3.4	3.4	NUM
cana-843	195	6	,	,	PUNCT
cana-843	195	7	satisfies	satisfy	VERB
cana-843	195	8	the	the	DET
cana-843	195	9	required	required	ADJ
cana-843	195	10	condition	condition	NOUN
cana-843	195	11	.	.	PUNCT
cana-843	196	1	communications	communication	NOUN
cana-843	196	2	on	on	ADP
cana-843	196	3	applied	apply	VERB
cana-843	196	4	nonlinear	nonlinear	ADJ
cana-843	196	5	analysis	analysis	NOUN
cana-843	196	6	issn	issn	NOUN
cana-843	196	7	:	:	PUNCT
cana-843	196	8	1074	1074	NUM
cana-843	196	9	-	-	PUNCT
cana-843	196	10	133x	133x	NUM
cana-843	196	11	vol	vol	NOUN
cana-843	196	12	31	31	NUM
cana-843	196	13	no	no	NOUN
cana-843	196	14	.	.	PUNCT
cana-843	197	1	4s	4s	NUM
cana-843	197	2	(	(	PUNCT
cana-843	197	3	2024	2024	NUM
cana-843	197	4	)	)	PUNCT
cana-843	197	5	226	226	NUM
cana-843	197	6	https://internationalpubls.com	https://internationalpubls.com	NOUN
cana-843	197	7	case	case	NOUN
cana-843	197	8	(	(	PUNCT
cana-843	197	9	iv	iv	X
cana-843	197	10	)	)	PUNCT
cana-843	197	11	0	0	PUNCT
cana-843	198	1	<	<	X
cana-843	198	2	𝑎	𝑎	X
cana-843	198	3	<	<	X
cana-843	198	4	𝑏.	𝑏.	NOUN
cana-843	198	5	consider	consider	VERB
cana-843	198	6	the	the	DET
cana-843	198	7	graph	graph	NOUN
cana-843	198	8	𝐺	𝐺	NOUN
cana-843	198	9	given	give	VERB
cana-843	198	10	in	in	ADP
cana-843	198	11	figure	figure	NOUN
cana-843	198	12	3.5	3.5	NUM
cana-843	198	13	.	.	PUNCT
cana-843	199	1	we	we	PRON
cana-843	199	2	first	first	ADV
cana-843	199	3	establish	establish	VERB
cana-843	199	4	that	that	SCONJ
cana-843	199	5	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	199	6	)	)	PUNCT
cana-843	199	7	=	=	PUNCT
cana-843	199	8	𝑎.	𝑎.	VERB
cana-843	199	9	the	the	DET
cana-843	199	10	set	set	NOUN
cana-843	199	11	of	of	ADP
cana-843	199	12	all	all	DET
cana-843	199	13	extreme	extreme	ADJ
cana-843	199	14	vertex	vertex	NOUN
cana-843	199	15	of	of	ADP
cana-843	199	16	𝐺	𝐺	PROPN
cana-843	199	17	is	be	AUX
cana-843	199	18	denoted	denote	VERB
cana-843	199	19	by	by	ADP
cana-843	199	20	𝑍	𝑍	PROPN
cana-843	199	21	=	=	SYM
cana-843	199	22	{	{	PUNCT
cana-843	199	23	𝑡	𝑡	PROPN
cana-843	199	24	,	,	PUNCT
cana-843	199	25	𝑤1	𝑤1	VERB
cana-843	199	26	,	,	PUNCT
cana-843	199	27	𝑤3	𝑤3	PROPN
cana-843	199	28	,	,	PUNCT
cana-843	199	29	𝑥1	𝑥1	NOUN
cana-843	199	30	,	,	PUNCT
cana-843	199	31	𝑥2	𝑥2	NOUN
cana-843	199	32	,	,	PUNCT
cana-843	199	33	…	…	PUNCT
cana-843	199	34	,	,	PUNCT
cana-843	199	35	𝑥𝑏−𝑎	𝑥𝑏−𝑎	NOUN
cana-843	199	36	,	,	PUNCT
cana-843	199	37	𝑦1	𝑦1	PROPN
cana-843	199	38	,	,	PUNCT
cana-843	199	39	𝑦2	𝑦2	NOUN
cana-843	199	40	,	,	PUNCT
cana-843	199	41	…	…	PUNCT
cana-843	199	42	,	,	PUNCT
cana-843	199	43	𝑦𝑏−𝑎	𝑦𝑏−𝑎	ADV
cana-843	199	44	}	}	PUNCT
cana-843	199	45	.	.	PUNCT
cana-843	200	1	thus	thus	ADV
cana-843	200	2	𝑍	𝑍	NOUN
cana-843	200	3	is	be	AUX
cana-843	200	4	a	a	DET
cana-843	200	5	subset	subset	NOUN
cana-843	200	6	of	of	ADP
cana-843	200	7	each	each	DET
cana-843	200	8	geodetic	geodetic	ADJ
cana-843	200	9	set	set	NOUN
cana-843	200	10	in	in	ADP
cana-843	200	11	𝐺	𝐺	PROPN
cana-843	200	12	,	,	PUNCT
cana-843	200	13	according	accord	VERB
cana-843	200	14	to	to	ADP
cana-843	200	15	theorem	theorem	NOUN
cana-843	200	16	3.4	3.4	NUM
cana-843	200	17	.	.	PUNCT
cana-843	201	1	𝐻𝑖	𝐻𝑖	ADJ
cana-843	201	2	:	:	PUNCT
cana-843	201	3	{	{	PUNCT
cana-843	201	4	𝑟𝑖	𝑟𝑖	PROPN
cana-843	201	5	,	,	PUNCT
cana-843	201	6	𝑠𝑖	𝑠𝑖	NOUN
cana-843	201	7	}	}	PUNCT
cana-843	201	8	(	(	PUNCT
cana-843	201	9	1	1	NUM
cana-843	201	10	≤	≤	NUM
cana-843	201	11	𝑖	𝑖	SYM
cana-843	201	12	≤	≤	NUM
cana-843	201	13	𝑎	𝑎	X
cana-843	201	14	)	)	PUNCT
cana-843	201	15	be	be	AUX
cana-843	201	16	given	give	VERB
cana-843	201	17	.	.	PUNCT
cana-843	202	1	every	every	DET
cana-843	202	2	geodetic	geodetic	ADJ
cana-843	202	3	set	set	NOUN
cana-843	202	4	of	of	ADP
cana-843	202	5	“	"	PUNCT
cana-843	202	6	𝐺	𝐺	PROPN
cana-843	202	7	has	have	VERB
cana-843	202	8	exactly	exactly	ADV
cana-843	202	9	one	one	NUM
cana-843	202	10	vertex	vertex	NOUN
cana-843	202	11	from	from	ADP
cana-843	202	12	each	each	DET
cana-843	202	13	𝐻𝑖	𝐻𝑖	PROPN
cana-843	202	14	(	(	PUNCT
cana-843	202	15	1	1	NUM
cana-843	202	16	≤	≤	NUM
cana-843	202	17	𝑖e	𝑖e	NOUN
cana-843	202	18	≤	≤	NUM
cana-843	202	19	𝑎	𝑎	NOUN
cana-843	202	20	)	)	PUNCT
cana-843	202	21	,	,	PUNCT
cana-843	202	22	as	as	SCONJ
cana-843	202	23	can	can	AUX
cana-843	202	24	be	be	AUX
cana-843	202	25	seen	see	VERB
cana-843	202	26	easily	easily	ADV
cana-843	202	27	.	.	PUNCT
cana-843	203	1	as	as	ADP
cana-843	203	2	a	a	DET
cana-843	203	3	result	result	NOUN
cana-843	203	4	,	,	PUNCT
cana-843	203	5	(	(	PUNCT
cana-843	203	6	𝐺	𝐺	NOUN
cana-843	203	7	)	)	PUNCT
cana-843	203	8	≥	≥	NOUN
cana-843	203	9	3	3	NUM
cana-843	204	1	+	+	CCONJ
cana-843	204	2	𝑏	𝑏	NOUN
cana-843	204	3	−	−	PROPN
cana-843	204	4	𝑎	𝑎	PROPN
cana-843	204	5	+	+	NOUN
cana-843	204	6	𝑏	𝑏	NOUN
cana-843	204	7	−	−	PROPN
cana-843	204	8	𝑎	𝑎	PROPN
cana-843	204	9	+	+	NUM
cana-843	204	10	𝑎	𝑎	PROPN
cana-843	204	11	=	=	NOUN
cana-843	204	12	2𝑏	2𝑏	NUM
cana-843	204	13	−	−	PROPN
cana-843	204	14	𝑎	𝑎	PUNCT
cana-843	204	15	+	+	NOUN
cana-843	204	16	3	3	X
cana-843	204	17	.	.	PUNCT
cana-843	204	18	let	let	VERB
cana-843	204	19	𝑆e	𝑆e	PROPN
cana-843	204	20	=	=	PUNCT
cana-843	204	21	𝑍	𝑍	PROPN
cana-843	204	22	∪	∪	ADJ
cana-843	204	23	{	{	PUNCT
cana-843	204	24	e𝑟1	e𝑟1	NOUN
cana-843	204	25	,	,	PUNCT
cana-843	204	26	e𝑟2	e𝑟2	PROPN
cana-843	204	27	,	,	PUNCT
cana-843	204	28	…	…	PUNCT
cana-843	204	29	,	,	PUNCT
cana-843	204	30	e𝑟𝑎	e𝑟𝑎	ADJ
cana-843	204	31	}	}	PUNCT
cana-843	204	32	.	.	PUNCT
cana-843	205	1	thus	thus	ADV
cana-843	205	2	,	,	PUNCT
cana-843	205	3	𝑆	𝑆	PROPN
cana-843	205	4	is	be	AUX
cana-843	205	5	a	a	DET
cana-843	205	6	geodetic	geodetic	ADJ
cana-843	205	7	set	set	NOUN
cana-843	205	8	of	of	ADP
cana-843	205	9	𝐺e	𝐺e	PROPN
cana-843	205	10	since	since	SCONJ
cana-843	205	11	𝐼[e𝑆	𝐼[e𝑆	PROPN
cana-843	205	12	]	]	X
cana-843	205	13	=	=	SYM
cana-843	205	14	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	205	15	)	)	PUNCT
cana-843	205	16	.	.	PUNCT
cana-843	206	1	consequently	consequently	ADV
cana-843	206	2	,	,	PUNCT
cana-843	206	3	𝑔(e𝐺	𝑔(e𝐺	PROPN
cana-843	206	4	)	)	PUNCT
cana-843	206	5	=	=	PUNCT
cana-843	206	6	2𝑏	2𝑏	NUM
cana-843	207	1	−	−	NOUN
cana-843	207	2	𝑎	𝑎	PUNCT
cana-843	207	3	+	+	NOUN
cana-843	207	4	3	3	NUM
cana-843	207	5	.	.	X
cana-843	207	6	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	207	7	)	)	PUNCT
cana-843	207	8	≤	≤	NUM
cana-843	207	9	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	207	10	)	)	PUNCT
cana-843	208	1	−	−	NOUN
cana-843	208	2	|𝑍|	|𝑍|	NOUN
cana-843	209	1	=	=	PUNCT
cana-843	209	2	2𝑏	2𝑏	NOUN
cana-843	210	1	−	−	PROPN
cana-843	211	1	𝑎	𝑎	X
cana-843	212	1	+	+	NOUN
cana-843	212	2	3	3	NUM
cana-843	212	3	−	−	NOUN
cana-843	212	4	(	(	PUNCT
cana-843	212	5	2𝑏	2𝑏	NUM
cana-843	212	6	−	−	NUM
cana-843	212	7	2𝑎	2𝑎	NUM
cana-843	212	8	+	+	CCONJ
cana-843	212	9	3	3	X
cana-843	212	10	)	)	PUNCT
cana-843	212	11	=	=	SYM
cana-843	212	12	𝑎	𝑎	NOUN
cana-843	212	13	,	,	PUNCT
cana-843	212	14	according	accord	VERB
cana-843	212	15	to	to	ADP
cana-843	212	16	the	the	DET
cana-843	212	17	theorem	theorem	NOUN
cana-843	212	18	.	.	PUNCT
cana-843	212	19	given	give	VERB
cana-843	212	20	that	that	DET
cana-843	212	21	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	212	22	)	)	PUNCT
cana-843	213	1	=	=	SYM
cana-843	213	2	2𝑏	2𝑏	NUM
cana-843	213	3	−	−	PROPN
cana-843	213	4	𝑎	𝑎	SYM
cana-843	213	5	+	+	NOUN
cana-843	213	6	3	3	NUM
cana-843	213	7	and	and	CCONJ
cana-843	213	8	that	that	SCONJ
cana-843	213	9	𝑍1	𝑍1	PROPN
cana-843	213	10	appears	appear	VERB
cana-843	213	11	in	in	ADP
cana-843	213	12	every	every	DET
cana-843	213	13	𝑔-set	𝑔-set	NOUN
cana-843	213	14	of	of	ADP
cana-843	213	15	𝐺	𝐺	PROPN
cana-843	213	16	,	,	PUNCT
cana-843	213	17	it	it	PRON
cana-843	213	18	is	be	AUX
cana-843	213	19	clear	clear	ADJ
cana-843	213	20	that	that	SCONJ
cana-843	213	21	every	every	DET
cana-843	213	22	𝑔-set	𝑔-set	NOUN
cana-843	213	23	of	of	ADP
cana-843	213	24	𝐺	𝐺	PROPN
cana-843	213	25	has	have	VERB
cana-843	213	26	the	the	DET
cana-843	213	27	form	form	NOUN
cana-843	213	28	𝑆	𝑆	PROPN
cana-843	213	29	=	=	SYM
cana-843	213	30	𝑍	𝑍	PROPN
cana-843	213	31	∪	∪	ADJ
cana-843	213	32	{	{	PUNCT
cana-843	213	33	e𝑐1	e𝑐1	NOUN
cana-843	213	34	,	,	PUNCT
cana-843	213	35	e𝑐2	e𝑐2	PROPN
cana-843	213	36	,	,	PUNCT
cana-843	213	37	…	…	PUNCT
cana-843	213	38	,	,	PUNCT
cana-843	213	39	e𝑐𝑎}where	e𝑐𝑎}where	ADV
cana-843	213	40	𝑐𝑖	𝑐𝑖	NOUN
cana-843	213	41	∈	∈	PROPN
cana-843	214	1	𝐻𝑖	𝐻𝑖	PROPN
cana-843	214	2	(	(	PUNCT
cana-843	214	3	1	1	NUM
cana-843	214	4	≤	≤	NUM
cana-843	214	5	𝑖	𝑖	SYM
cana-843	214	6	≤	≤	NUM
cana-843	214	7	e𝑎	e𝑎	NOUN
cana-843	214	8	)	)	PUNCT
cana-843	214	9	.	.	PUNCT
cana-843	215	1	given	give	VERB
cana-843	215	2	|𝑇|	|𝑇|	NOUN
cana-843	215	3	<	<	X
cana-843	215	4	𝑎	𝑎	NOUN
cana-843	215	5	,	,	PUNCT
cana-843	215	6	let	let	VERB
cana-843	215	7	e𝑇	e𝑇	ADJ
cana-843	215	8	be	be	AUX
cana-843	215	9	any	any	DET
cana-843	215	10	suitable	suitable	ADJ
cana-843	215	11	subset	subset	NOUN
cana-843	215	12	of	of	ADP
cana-843	215	13	𝑆.	𝑆.	PROPN
cana-843	215	14	after	after	ADP
cana-843	215	15	that	that	PRON
cana-843	215	16	,	,	PUNCT
cana-843	215	17	𝑐𝑗	𝑐𝑗	CCONJ
cana-843	215	18	(	(	PUNCT
cana-843	215	19	1	1	NUM
cana-843	215	20	≤	≤	NUM
cana-843	215	21	𝑗	𝑗	PRON
cana-843	215	22	≤	≤	NUM
cana-843	215	23	𝑎	𝑎	NOUN
cana-843	215	24	)	)	PUNCT
cana-843	215	25	”	"	PUNCT
cana-843	215	26	is	be	AUX
cana-843	215	27	a	a	DET
cana-843	215	28	vertex	vertex	NOUN
cana-843	215	29	such	such	ADJ
cana-843	215	30	that	that	DET
cana-843	215	31	e𝑐𝑗	e𝑐𝑗	PROPN
cana-843	215	32	∉	∉	PROPN
cana-843	215	33	𝑇.	𝑇.	PROPN
cana-843	215	34	assume	assume	VERB
cana-843	215	35	that	that	SCONJ
cana-843	215	36	e𝑏	e𝑏	PROPN
cana-843	215	37	𝑗	𝑗	INTJ
cana-843	215	38	,	,	PUNCT
cana-843	215	39	a	a	DET
cana-843	215	40	vertex	vertex	NOUN
cana-843	215	41	of	of	ADP
cana-843	215	42	𝐻𝑗	𝐻𝑗	PROPN
cana-843	215	43	,	,	PUNCT
cana-843	215	44	is	be	AUX
cana-843	215	45	different	different	ADJ
cana-843	215	46	from	from	ADP
cana-843	215	47	𝑐𝑗.	𝑐𝑗.	NOUN
cana-843	215	48	subsequently	subsequently	ADV
cana-843	215	49	,	,	PUNCT
cana-843	215	50	𝑆1	𝑆1	NOUN
cana-843	215	51	=	=	SYM
cana-843	215	52	(	(	PUNCT
cana-843	215	53	𝑆	𝑆	PROPN
cana-843	215	54	−	−	PROPN
cana-843	215	55	{	{	PUNCT
cana-843	215	56	𝑐𝑗	𝑐𝑗	NOUN
cana-843	215	57	}	}	PUNCT
cana-843	215	58	)	)	PUNCT
cana-843	215	59	∪	∪	SCONJ
cana-843	215	60	{	{	PUNCT
cana-843	215	61	𝑏𝑗	𝑏𝑗	NOUN
cana-843	215	62	}	}	PUNCT
cana-843	215	63	is	be	AUX
cana-843	215	64	a	a	DET
cana-843	215	65	𝑔-set	𝑔-set	NOUN
cana-843	215	66	that	that	PRON
cana-843	215	67	correctly	correctly	ADV
cana-843	215	68	contains	contain	VERB
cana-843	215	69	e𝑇.	e𝑇.	ADJ
cana-843	215	70	such	such	ADJ
cana-843	215	71	that	that	SCONJ
cana-843	215	72	e𝑇	e𝑇	PROPN
cana-843	215	73	is	be	AUX
cana-843	215	74	not	not	PART
cana-843	215	75	a	a	DET
cana-843	215	76	forced	force	VERB
cana-843	215	77	subset	subset	NOUN
cana-843	215	78	of	of	ADP
cana-843	215	79	𝑆	𝑆	PROPN
cana-843	215	80	as	as	ADP
cana-843	215	81	a	a	DET
cana-843	215	82	result	result	NOUN
cana-843	215	83	.	.	PUNCT
cana-843	216	1	for	for	ADP
cana-843	216	2	every	every	DET
cana-843	216	3	smallest	small	ADJ
cana-843	216	4	𝑔-set	𝑔-set	NOUN
cana-843	216	5	of	of	ADP
cana-843	216	6	𝐺	𝐺	PROPN
cana-843	216	7	,	,	PUNCT
cana-843	216	8	this	this	PRON
cana-843	216	9	holds	hold	VERB
cana-843	216	10	true	true	ADJ
cana-843	216	11	.	.	PUNCT
cana-843	217	1	hence	hence	ADV
cana-843	217	2	,	,	PUNCT
cana-843	217	3	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	217	4	)	)	PUNCT
cana-843	217	5	=	=	PUNCT
cana-843	218	1	𝑎.	𝑎.	NOUN
cana-843	218	2	figure	figure	NOUN
cana-843	218	3	3.5	3.5	NUM
cana-843	218	4	we	we	PRON
cana-843	218	5	then	then	ADV
cana-843	218	6	demonstrate	demonstrate	VERB
cana-843	218	7	that	that	SCONJ
cana-843	218	8	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	218	9	)	)	PUNCT
cana-843	219	1	=	=	VERB
cana-843	219	2	𝑏.	𝑏.	ADV
cana-843	219	3	let	let	VERB
cana-843	219	4	𝑍1	𝑍1	NOUN
cana-843	219	5	=	=	PUNCT
cana-843	219	6	{	{	PUNCT
cana-843	219	7	𝑡	𝑡	NOUN
cana-843	219	8	}	}	PUNCT
cana-843	219	9	represent	represent	VERB
cana-843	219	10	𝐺	𝐺	PROPN
cana-843	219	11	's	's	PART
cana-843	219	12	end	end	NOUN
cana-843	219	13	vertex	vertex	NOUN
cana-843	219	14	.	.	PUNCT
cana-843	220	1	such	such	ADJ
cana-843	220	2	that	that	DET
cana-843	220	3	𝑍	𝑍	PROPN
cana-843	220	4	is	be	AUX
cana-843	220	5	a	a	DET
cana-843	220	6	subset	subset	NOUN
cana-843	220	7	of	of	ADP
cana-843	220	8	every	every	DET
cana-843	220	9	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	220	10	in	in	ADP
cana-843	220	11	𝐺	𝐺	PROPN
cana-843	220	12	according	accord	VERB
cana-843	220	13	to	to	ADP
cana-843	220	14	theorem	theorem	ADJ
cana-843	220	15	1.1	1.1	NUM
cana-843	220	16	.	.	PUNCT
cana-843	221	1	𝑄𝑗	𝑄𝑗	PROPN
cana-843	221	2	:	:	PUNCT
cana-843	221	3	{	{	PUNCT
cana-843	221	4	𝑥𝑗	𝑥𝑗	PROPN
cana-843	221	5	,	,	PUNCT
cana-843	221	6	𝑦𝑗	𝑦𝑗	PROPN
cana-843	221	7	}	}	PUNCT
cana-843	221	8	(	(	PUNCT
cana-843	221	9	1	1	NUM
cana-843	221	10	≤	≤	NUM
cana-843	221	11	𝑗	𝑗	PRON
cana-843	221	12	≤	≤	NOUN
cana-843	221	13	𝑏	𝑏	DET
cana-843	221	14	−	−	PROPN
cana-843	221	15	𝑎	𝑎	NOUN
cana-843	221	16	)	)	PUNCT
cana-843	221	17	be	be	AUX
cana-843	221	18	given	give	VERB
cana-843	221	19	.	.	PUNCT
cana-843	222	1	every	every	DET
cana-843	222	2	circular	circular	ADJ
cana-843	222	3	set	set	NOUN
cana-843	222	4	of	of	ADP
cana-843	222	5	𝐺	𝐺	PROPN
cana-843	222	6	has	have	VERB
cana-843	222	7	exactly	exactly	ADV
cana-843	222	8	one	one	NUM
cana-843	222	9	vertex	vertex	NOUN
cana-843	222	10	from	from	ADP
cana-843	222	11	every	every	DET
cana-843	222	12	𝑄𝑗	𝑄𝑗	PROPN
cana-843	222	13	(	(	PUNCT
cana-843	222	14	1	1	NUM
cana-843	222	15	≤	≤	NUM
cana-843	222	16	𝑗	𝑗	PRON
cana-843	222	17	≤	≤	NOUN
cana-843	222	18	𝑏	𝑏	DET
cana-843	222	19	−	−	PROPN
cana-843	222	20	𝑎	𝑎	NOUN
cana-843	222	21	)	)	PUNCT
cana-843	222	22	and	and	CCONJ
cana-843	222	23	exactly	exactly	ADV
cana-843	222	24	one	one	NUM
cana-843	222	25	vertex	vertex	NOUN
cana-843	222	26	from	from	ADP
cana-843	222	27	every	every	DET
cana-843	222	28	𝐻𝑖	𝐻𝑖	PROPN
cana-843	222	29	(	(	PUNCT
cana-843	222	30	1	1	NUM
cana-843	222	31	≤	≤	NUM
cana-843	222	32	𝑖	𝑖	SYM
cana-843	222	33	≤	≤	NUM
cana-843	222	34	𝑎	𝑎	NOUN
cana-843	222	35	)	)	PUNCT
cana-843	222	36	,	,	PUNCT
cana-843	222	37	as	as	SCONJ
cana-843	222	38	can	can	AUX
cana-843	222	39	be	be	AUX
cana-843	222	40	seen	see	VERB
cana-843	222	41	easily	easily	ADV
cana-843	222	42	.	.	PUNCT
cana-843	223	1	therefore	therefore	ADV
cana-843	223	2	,	,	PUNCT
cana-843	223	3	𝑐𝑟(𝐺	𝑐𝑟(𝐺	ADV
cana-843	223	4	)	)	PUNCT
cana-843	223	5	≥	≥	NOUN
cana-843	223	6	1	1	NUM
cana-843	224	1	+	+	CCONJ
cana-843	225	1	𝑎	𝑎	X
cana-843	225	2	+	+	NOUN
cana-843	225	3	𝑏	𝑏	NOUN
cana-843	225	4	−	−	PROPN
cana-843	225	5	𝑎	𝑎	PROPN
cana-843	225	6	=	=	SYM
cana-843	225	7	𝑏	𝑏	NOUN
cana-843	225	8	+	+	NOUN
cana-843	225	9	1	1	X
cana-843	225	10	.	.	X
cana-843	225	11	assume	assume	VERB
cana-843	225	12	that	that	SCONJ
cana-843	225	13	𝑆	𝑆	PROPN
cana-843	225	14	=	=	SYM
cana-843	225	15	𝑍	𝑍	PROPN
cana-843	225	16	∪	∪	X
cana-843	225	17	{	{	PUNCT
cana-843	225	18	𝑟1	𝑟1	NOUN
cana-843	225	19	,	,	PUNCT
cana-843	225	20	𝑟2	𝑟2	NOUN
cana-843	225	21	,	,	PUNCT
cana-843	225	22	…	…	PUNCT
cana-843	225	23	,	,	PUNCT
cana-843	225	24	𝑟𝑎	𝑟𝑎	X
cana-843	225	25	}	}	PUNCT
cana-843	225	26	∪	∪	ADJ
cana-843	225	27	{	{	PUNCT
cana-843	225	28	𝑥1	𝑥1	NOUN
cana-843	225	29	,	,	PUNCT
cana-843	225	30	𝑥2	𝑥2	NOUN
cana-843	225	31	,	,	PUNCT
cana-843	225	32	…	…	PUNCT
cana-843	225	33	,	,	PUNCT
cana-843	225	34	𝑥𝑏−𝑎	𝑥𝑏−𝑎	ADV
cana-843	225	35	}	}	PUNCT
cana-843	225	36	.	.	PUNCT
cana-843	226	1	as	as	ADP
cana-843	226	2	a	a	DET
cana-843	226	3	result	result	NOUN
cana-843	226	4	,	,	PUNCT
cana-843	226	5	𝑆	𝑆	PROPN
cana-843	226	6	is	be	AUX
cana-843	226	7	a	a	DET
cana-843	226	8	circular	circular	ADJ
cana-843	226	9	set	set	NOUN
cana-843	226	10	of	of	ADP
cana-843	226	11	𝐺	𝐺	PROPN
cana-843	226	12	since	since	SCONJ
cana-843	226	13	𝐼[𝑆	𝐼[𝑆	NOUN
cana-843	226	14	]	]	X
cana-843	226	15	=	=	SYM
cana-843	226	16	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	226	17	)	)	PUNCT
cana-843	226	18	.	.	PUNCT
cana-843	227	1	consequently	consequently	ADV
cana-843	227	2	,	,	PUNCT
cana-843	227	3	𝑐𝑟(𝐺	𝑐𝑟(𝐺	ADV
cana-843	227	4	)	)	PUNCT
cana-843	227	5	=	=	SYM
cana-843	228	1	𝑏	𝑏	PROPN
cana-843	228	2	+	+	NOUN
cana-843	228	3	1	1	NUM
cana-843	228	4	.	.	X
cana-843	228	5	therefore	therefore	ADV
cana-843	228	6	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	228	7	)	)	PUNCT
cana-843	228	8	≤	≤	NOUN
cana-843	228	9	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	228	10	)	)	PUNCT
cana-843	228	11	−	−	NOUN
cana-843	228	12	|𝑍|	|𝑍|	NOUN
cana-843	228	13	=	=	PUNCT
cana-843	229	1	𝑏	𝑏	NOUN
cana-843	229	2	+	+	NOUN
cana-843	229	3	1	1	NUM
cana-843	229	4	−	−	NUM
cana-843	229	5	1	1	NUM
cana-843	229	6	=	=	SYM
cana-843	229	7	𝑏	𝑏	NOUN
cana-843	229	8	,	,	PUNCT
cana-843	229	9	according	accord	VERB
cana-843	229	10	to	to	ADP
cana-843	229	11	the	the	DET
cana-843	229	12	theorem	theorem	NOUN
cana-843	229	13	.	.	PUNCT
cana-843	230	1	every	every	DET
cana-843	230	2	circular	circular	ADJ
cana-843	230	3	set	set	NOUN
cana-843	230	4	of	of	ADP
cana-843	230	5	𝐺	𝐺	PROPN
cana-843	230	6	,	,	PUNCT
cana-843	230	7	of	of	ADP
cana-843	230	8	which	which	PRON
cana-843	230	9	𝑍	𝑍	VERB
cana-843	230	10	is	be	AUX
cana-843	230	11	a	a	DET
cana-843	230	12	subset	subset	NOUN
cana-843	230	13	,	,	PUNCT
cana-843	230	14	has	have	AUX
cana-843	230	15	the	the	DET
cana-843	230	16	form	form	NOUN
cana-843	230	17	𝑊1	𝑊1	PROPN
cana-843	230	18	=	=	SYM
cana-843	230	19	𝑍	𝑍	PROPN
cana-843	230	20	∪	∪	ADJ
cana-843	230	21	{	{	PUNCT
cana-843	230	22	𝑐1	𝑐1	NOUN
cana-843	230	23	,	,	PUNCT
cana-843	230	24	𝑐2	𝑐2	NOUN
cana-843	230	25	,	,	PUNCT
cana-843	230	26	…	…	PUNCT
cana-843	230	27	,	,	PUNCT
cana-843	230	28	𝑐𝑎	𝑐𝑎	NOUN
cana-843	230	29	}	}	PUNCT
cana-843	230	30	∪	∪	ADJ
cana-843	230	31	{	{	PUNCT
cana-843	230	32	𝑑1	𝑑1	NOUN
cana-843	230	33	,	,	PUNCT
cana-843	230	34	𝑑2	𝑑2	NOUN
cana-843	230	35	,	,	PUNCT
cana-843	230	36	…	…	PUNCT
cana-843	230	37	,	,	PUNCT
cana-843	230	38	𝑑𝑏−𝑎	𝑑𝑏−𝑎	PROPN
cana-843	230	39	}	}	PUNCT
cana-843	230	40	,	,	PUNCT
cana-843	230	41	where	where	SCONJ
cana-843	230	42	𝑑𝑗	𝑑𝑗	PROPN
cana-843	230	43	∈	∈	PROPN
cana-843	230	44	𝑄𝑗	𝑄𝑗	PROPN
cana-843	230	45	(	(	PUNCT
cana-843	230	46	1	1	NUM
cana-843	230	47	≤	≤	NUM
cana-843	230	48	𝑗	𝑗	PRON
cana-843	230	49	≤	≤	NOUN
cana-843	230	50	𝑏	𝑏	DET
cana-843	230	51	−	−	PROPN
cana-843	230	52	𝑎	𝑎	NOUN
cana-843	230	53	)	)	PUNCT
cana-843	230	54	and	and	CCONJ
cana-843	230	55	𝑐𝑖	𝑐𝑖	NOUN
cana-843	230	56	∈	∈	PROPN
cana-843	230	57	𝐻𝑖	𝐻𝑖	PROPN
cana-843	230	58	(	(	PUNCT
cana-843	230	59	1	1	NUM
cana-843	230	60	≤	≤	NUM
cana-843	230	61	𝑖	𝑖	SYM
cana-843	230	62	≤	≤	NUM
cana-843	230	63	𝑎	𝑎	X
cana-843	230	64	)	)	PUNCT
cana-843	230	65	.	.	PUNCT
cana-843	231	1	given	give	VERB
cana-843	231	2	|𝑇|	|𝑇|	NOUN
cana-843	231	3	<	<	X
cana-843	231	4	𝑏	𝑏	NOUN
cana-843	231	5	,	,	PUNCT
cana-843	231	6	let	let	VERB
cana-843	231	7	𝑇	𝑇	PROPN
cana-843	231	8	be	be	AUX
cana-843	231	9	any	any	DET
cana-843	231	10	proper	proper	ADJ
cana-843	231	11	subset	subset	NOUN
cana-843	231	12	of	of	ADP
cana-843	231	13	𝑊1	𝑊1	PROPN
cana-843	231	14	.	.	PUNCT
cana-843	232	1	after	after	ADP
cana-843	232	2	that	that	PRON
cana-843	232	3	,	,	PUNCT
cana-843	232	4	𝑐𝑗	𝑐𝑗	INTJ
cana-843	232	5	,	,	PUNCT
cana-843	232	6	𝑑𝑗	𝑑𝑗	PROPN
cana-843	232	7	∉	∉	PROPN
cana-843	232	8	𝑇	𝑇	PROPN
cana-843	232	9	since	since	SCONJ
cana-843	232	10	there	there	PRON
cana-843	232	11	are	be	VERB
cana-843	232	12	vertices	vertex	NOUN
cana-843	232	13	𝑐𝑗	𝑐𝑗	NOUN
cana-843	232	14	∈	∈	PROPN
cana-843	232	15	𝐻𝑖	𝐻𝑖	PROPN
cana-843	232	16	and	and	CCONJ
cana-843	232	17	𝑑𝑗	𝑑𝑗	ADP
cana-843	232	18	∈	∈	PROPN
cana-843	233	1	𝑄𝑗.	𝑄𝑗.	PROPN
cana-843	233	2	assume	assume	VERB
cana-843	233	3	that	that	SCONJ
cana-843	233	4	𝑓𝑗	𝑓𝑗	ADJ
cana-843	233	5	is	be	AUX
cana-843	233	6	a	a	DET
cana-843	233	7	vertex	vertex	NOUN
cana-843	233	8	of	of	ADP
cana-843	233	9	𝑄𝑗	𝑄𝑗	PROPN
cana-843	233	10	that	that	PRON
cana-843	233	11	is	be	AUX
cana-843	233	12	separate	separate	ADJ
cana-843	233	13	from	from	ADP
cana-843	233	14	𝑑𝑗	𝑑𝑗	PRON
cana-843	233	15	and	and	CCONJ
cana-843	233	16	that	that	SCONJ
cana-843	233	17	𝑒𝑖	𝑒𝑖	NOUN
cana-843	233	18	is	be	AUX
cana-843	233	19	a	a	DET
cana-843	233	20	vertex	vertex	NOUN
cana-843	233	21	of	of	ADP
cana-843	233	22	𝐻𝑖	𝐻𝑖	PROPN
cana-843	233	23	apart	apart	ADV
cana-843	233	24	from	from	ADP
cana-843	233	25	𝑐𝑖.	𝑐𝑖.	PROPN
cana-843	233	26	𝑊2	𝑊2	PROPN
cana-843	233	27	=	=	SYM
cana-843	233	28	communications	communication	NOUN
cana-843	233	29	on	on	ADP
cana-843	233	30	applied	apply	VERB
cana-843	233	31	nonlinear	nonlinear	ADJ
cana-843	233	32	analysis	analysis	NOUN
cana-843	233	33	issn	issn	NOUN
cana-843	233	34	:	:	PUNCT
cana-843	233	35	1074	1074	NUM
cana-843	233	36	-	-	PUNCT
cana-843	233	37	133x	133x	NUM
cana-843	233	38	vol	vol	NOUN
cana-843	233	39	31	31	NUM
cana-843	233	40	no	no	NOUN
cana-843	233	41	.	.	PUNCT
cana-843	234	1	4s	4s	NUM
cana-843	234	2	(	(	PUNCT
cana-843	234	3	2024	2024	NUM
cana-843	234	4	)	)	PUNCT
cana-843	234	5	227	227	NUM
cana-843	234	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	234	7	(	(	PUNCT
cana-843	234	8	𝑊1	𝑊1	PROPN
cana-843	234	9	−	−	PROPN
cana-843	234	10	{	{	PUNCT
cana-843	234	11	𝑐𝑗	𝑐𝑗	PROPN
cana-843	234	12	,	,	PUNCT
cana-843	234	13	𝑑𝑗	𝑑𝑗	NOUN
cana-843	234	14	}	}	PUNCT
cana-843	234	15	)	)	PUNCT
cana-843	234	16	∪	∪	ADP
cana-843	234	17	{	{	PUNCT
cana-843	234	18	𝑒𝑖	𝑒𝑖	NOUN
cana-843	234	19	,	,	PUNCT
cana-843	234	20	𝑓𝑗	𝑓𝑗	ADJ
cana-843	234	21	}	}	PUNCT
cana-843	234	22	is	be	AUX
cana-843	234	23	a	a	DET
cana-843	234	24	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	234	25	that	that	PRON
cana-843	234	26	correctly	correctly	ADV
cana-843	234	27	contains	contain	VERB
cana-843	234	28	𝑇	𝑇	PROPN
cana-843	234	29	in	in	ADP
cana-843	234	30	this	this	DET
cana-843	234	31	case	case	NOUN
cana-843	234	32	.	.	PUNCT
cana-843	235	1	such	such	ADJ
cana-843	235	2	that	that	SCONJ
cana-843	235	3	𝑇	𝑇	PROPN
cana-843	235	4	is	be	AUX
cana-843	235	5	not	not	PART
cana-843	235	6	a	a	DET
cana-843	235	7	forced	force	VERB
cana-843	235	8	subset	subset	NOUN
cana-843	235	9	of	of	ADP
cana-843	235	10	𝑊2	𝑊2	PROPN
cana-843	235	11	as	as	ADP
cana-843	235	12	a	a	DET
cana-843	235	13	result	result	NOUN
cana-843	235	14	.	.	PUNCT
cana-843	236	1	for	for	ADP
cana-843	236	2	every	every	DET
cana-843	236	3	minimum	minimum	ADJ
cana-843	236	4	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	236	5	of	of	ADP
cana-843	236	6	𝐺	𝐺	PROPN
cana-843	236	7	,	,	PUNCT
cana-843	236	8	this	this	PRON
cana-843	236	9	is	be	AUX
cana-843	236	10	true	true	ADJ
cana-843	236	11	.	.	PUNCT
cana-843	237	1	thus	thus	ADV
cana-843	237	2	,	,	PUNCT
cana-843	237	3	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	237	4	)	)	PUNCT
cana-843	237	5	=	=	SYM
cana-843	237	6	𝑏.	𝑏.	NOUN
cana-843	237	7	case	case	NOUN
cana-843	237	8	(	(	PUNCT
cana-843	237	9	v	v	NOUN
cana-843	237	10	)	)	PUNCT
cana-843	237	11	0	0	PUNCT
cana-843	238	1	<	<	X
cana-843	238	2	𝑏	𝑏	X
cana-843	238	3	<	<	X
cana-843	238	4	𝑎.	𝑎.	PROPN
cana-843	238	5	first	first	ADV
cana-843	238	6	,	,	PUNCT
cana-843	238	7	we	we	PRON
cana-843	238	8	establish	establish	VERB
cana-843	238	9	that	that	SCONJ
cana-843	238	10	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	238	11	)	)	PUNCT
cana-843	239	1	=	=	VERB
cana-843	239	2	𝑏.	𝑏.	NOUN
cana-843	239	3	assume	assume	VERB
cana-843	239	4	𝑍	𝑍	PROPN
cana-843	239	5	=	=	SYM
cana-843	239	6	{	{	PUNCT
cana-843	239	7	𝑡1	𝑡1	NOUN
cana-843	239	8	,	,	PUNCT
cana-843	239	9	w	w	NOUN
cana-843	239	10	}	}	PUNCT
cana-843	239	11	.	.	PUNCT
cana-843	240	1	thus	thus	ADV
cana-843	240	2	𝑍	𝑍	NOUN
cana-843	240	3	is	be	AUX
cana-843	240	4	therefore	therefore	ADV
cana-843	240	5	a	a	DET
cana-843	240	6	subset	subset	NOUN
cana-843	240	7	of	of	ADP
cana-843	240	8	𝐺	𝐺	PROPN
cana-843	240	9	's	's	PART
cana-843	240	10	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	240	11	.	.	PUNCT
cana-843	241	1	assume	assume	VERB
cana-843	241	2	“	"	PUNCT
cana-843	241	3	𝐻𝑖e	𝐻𝑖e	PROPN
cana-843	241	4	:	:	PUNCT
cana-843	241	5	{	{	PUNCT
cana-843	241	6	𝑟𝑖	𝑟𝑖	PART
cana-843	241	7	e	e	NOUN
cana-843	241	8	,	,	PUNCT
cana-843	241	9	e𝑠𝑖	e𝑠𝑖	PROPN
cana-843	241	10	}	}	PUNCT
cana-843	241	11	(	(	PUNCT
cana-843	241	12	1	1	NUM
cana-843	241	13	≤	≤	NUM
cana-843	241	14	𝑖e	𝑖e	NOUN
cana-843	241	15	≤	≤	ADJ
cana-843	241	16	𝑏)be	𝑏)be	PROPN
cana-843	241	17	given	give	VERB
cana-843	241	18	.	.	PUNCT
cana-843	242	1	it	it	PRON
cana-843	242	2	is	be	AUX
cana-843	242	3	simple	simple	ADJ
cana-843	242	4	to	to	PART
cana-843	242	5	see	see	VERB
cana-843	242	6	that	that	SCONJ
cana-843	242	7	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	242	8	)	)	PUNCT
cana-843	242	9	≥	≥	NOUN
cana-843	243	1	𝑏	𝑏	NOUN
cana-843	243	2	+	+	NOUN
cana-843	243	3	2	2	NUM
cana-843	243	4	as	as	SCONJ
cana-843	243	5	every	every	DET
cana-843	243	6	circular	circular	ADJ
cana-843	243	7	set	set	NOUN
cana-843	243	8	of	of	ADP
cana-843	243	9	𝐺	𝐺	PROPN
cana-843	243	10	has	have	VERB
cana-843	243	11	at	at	ADV
cana-843	243	12	least	least	ADV
cana-843	243	13	one	one	NUM
cana-843	243	14	vertex	vertex	NOUN
cana-843	243	15	from	from	ADP
cana-843	243	16	each	each	DET
cana-843	243	17	𝐻𝑖(1	𝐻𝑖(1	ADJ
cana-843	243	18	≤	≤	NUM
cana-843	243	19	𝑖	𝑖	SYM
cana-843	243	20	≤	≤	NUM
cana-843	243	21	𝑎	𝑎	X
cana-843	243	22	)	)	PUNCT
cana-843	243	23	.	.	PUNCT
cana-843	244	1	consider	consider	VERB
cana-843	244	2	𝑆	𝑆	PROPN
cana-843	244	3	=	=	PRON
cana-843	244	4	𝑍1	𝑍1	PROPN
cana-843	244	5	∪	∪	X
cana-843	244	6	{	{	PUNCT
cana-843	244	7	e𝑟1	e𝑟1	NOUN
cana-843	244	8	,	,	PUNCT
cana-843	244	9	e𝑟2	e𝑟2	PROPN
cana-843	244	10	,	,	PUNCT
cana-843	244	11	…	…	PUNCT
cana-843	244	12	,	,	PUNCT
cana-843	244	13	e𝑟𝑏	e𝑟𝑏	NOUN
cana-843	244	14	}	}	PUNCT
cana-843	244	15	.	.	PUNCT
cana-843	245	1	consequently	consequently	ADV
cana-843	245	2	,	,	PUNCT
cana-843	245	3	𝑆	𝑆	PROPN
cana-843	245	4	is	be	AUX
cana-843	245	5	a	a	DET
cana-843	245	6	circular	circular	ADJ
cana-843	245	7	set	set	NOUN
cana-843	245	8	of	of	ADP
cana-843	245	9	e𝐺	e𝐺	PROPN
cana-843	245	10	since	since	SCONJ
cana-843	245	11	𝐼[e𝑆	𝐼[e𝑆	PROPN
cana-843	245	12	]	]	X
cana-843	245	13	=	=	SYM
cana-843	245	14	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	245	15	)	)	PUNCT
cana-843	245	16	,	,	PUNCT
cana-843	245	17	and	and	CCONJ
cana-843	245	18	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NUM
cana-843	245	19	)	)	PUNCT
cana-843	245	20	=	=	SYM
cana-843	245	21	𝑏	𝑏	PROPN
cana-843	245	22	+	+	NOUN
cana-843	245	23	2	2	NUM
cana-843	245	24	.	.	PUNCT
cana-843	245	25	given	give	VERB
cana-843	245	26	that	that	PRON
cana-843	245	27	e𝑍	e𝑍	NOUN
cana-843	245	28	is	be	AUX
cana-843	245	29	a	a	DET
cana-843	245	30	subset	subset	NOUN
cana-843	245	31	of	of	ADP
cana-843	245	32	each	each	DET
cana-843	245	33	𝐺	𝐺	PROPN
cana-843	245	34	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	245	35	.	.	PUNCT
cana-843	246	1	according	accord	VERB
cana-843	246	2	to	to	ADP
cana-843	246	3	theorem	theorem	ADJ
cana-843	246	4	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	246	5	)	)	PUNCT
cana-843	246	6	≤	≤	NOUN
cana-843	246	7	𝑐𝑟(𝐺	𝑐𝑟(𝐺	NOUN
cana-843	246	8	)	)	PUNCT
cana-843	246	9	−	−	NOUN
cana-843	246	10	|𝑍|	|𝑍|	NOUN
cana-843	246	11	=	=	PUNCT
cana-843	247	1	+2	+2	PRON
cana-843	247	2	−	−	PROPN
cana-843	247	3	2	2	NUM
cana-843	247	4	=	=	SYM
cana-843	247	5	e𝑏.	e𝑏.	NOUN
cana-843	247	6	thus	thus	ADV
cana-843	247	7	,	,	PUNCT
cana-843	247	8	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	247	9	)	)	PUNCT
cana-843	247	10	≤	≤	NOUN
cana-843	247	11	𝑏.	𝑏.	VERB
cana-843	247	12	it	it	PRON
cana-843	247	13	is	be	AUX
cana-843	247	14	clear	clear	ADJ
cana-843	247	15	that	that	SCONJ
cana-843	247	16	every	every	DET
cana-843	247	17	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	247	18	of	of	ADP
cana-843	247	19	𝐺	𝐺	PROPN
cana-843	247	20	is	be	AUX
cana-843	247	21	of	of	ADP
cana-843	247	22	the	the	DET
cana-843	247	23	type	type	NOUN
cana-843	247	24	𝑆	𝑆	PROPN
cana-843	247	25	=	=	SYM
cana-843	247	26	e𝑍1	e𝑍1	PROPN
cana-843	247	27	∪	∪	ADJ
cana-843	247	28	{	{	PUNCT
cana-843	247	29	𝑐1	𝑐1	NOUN
cana-843	247	30	,	,	PUNCT
cana-843	247	31	e𝑐2	e𝑐2	PROPN
cana-843	247	32	,	,	PUNCT
cana-843	247	33	…	…	PUNCT
cana-843	247	34	,	,	PUNCT
cana-843	247	35	e𝑐𝑎	e𝑐𝑎	PROPN
cana-843	247	36	}	}	PUNCT
cana-843	247	37	,	,	PUNCT
cana-843	247	38	where	where	SCONJ
cana-843	247	39	𝑐𝑖	𝑐𝑖	PROPN
cana-843	247	40	∈	∈	PROPN
cana-843	247	41	𝐻𝑖(1	𝐻𝑖(1	PROPN
cana-843	247	42	≤	≤	NUM
cana-843	247	43	𝑖	𝑖	SYM
cana-843	247	44	≤	≤	NUM
cana-843	248	1	𝑎	𝑎	NOUN
cana-843	248	2	)	)	PUNCT
cana-843	248	3	,	,	PUNCT
cana-843	248	4	since	since	SCONJ
cana-843	248	5	𝑐𝑟(𝐺	𝑐𝑟(𝐺	ADV
cana-843	248	6	)	)	PUNCT
cana-843	249	1	=	=	SYM
cana-843	249	2	𝑏	𝑏	PROPN
cana-843	250	1	+	+	CCONJ
cana-843	251	1	2	2	NUM
cana-843	252	1	and	and	CCONJ
cana-843	252	2	every	every	DET
cana-843	252	3	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	252	4	of	of	ADP
cana-843	252	5	𝐺	𝐺	PROPN
cana-843	252	6	contains	contain	VERB
cana-843	252	7	𝑍.	𝑍.	PROPN
cana-843	252	8	given	give	VERB
cana-843	252	9	|𝑇|	|𝑇|	PROPN
cana-843	252	10	<	<	X
cana-843	252	11	𝑎	𝑎	NOUN
cana-843	252	12	,	,	PUNCT
cana-843	252	13	let	let	VERB
cana-843	252	14	𝑇	𝑇	PROPN
cana-843	252	15	be	be	AUX
cana-843	252	16	any	any	DET
cana-843	252	17	proper	proper	ADJ
cana-843	252	18	subset	subset	NOUN
cana-843	252	19	of	of	ADP
cana-843	252	20	𝑆.	𝑆.	PROPN
cana-843	252	21	after	after	ADP
cana-843	252	22	that	that	PRON
cana-843	252	23	,	,	PUNCT
cana-843	252	24	e𝑐𝑗	e𝑐𝑗	INTJ
cana-843	252	25	(	(	PUNCT
cana-843	252	26	1	1	NUM
cana-843	252	27	≤	≤	NUM
cana-843	252	28	e𝑗	e𝑗	ADP
cana-843	252	29	≤	≤	ADJ
cana-843	252	30	e𝑎	e𝑎	NOUN
cana-843	252	31	)	)	PUNCT
cana-843	252	32	is	be	AUX
cana-843	252	33	a	a	DET
cana-843	252	34	vertex	vertex	NOUN
cana-843	252	35	such	such	ADJ
cana-843	252	36	that	that	SCONJ
cana-843	252	37	e𝑐	e𝑐	PRON
cana-843	252	38	𝑗	𝑗	PRON
cana-843	252	39	∉	∉	X
cana-843	252	40	e𝑇.	e𝑇.	ADJ
cana-843	252	41	assume	assume	NOUN
cana-843	252	42	that	that	SCONJ
cana-843	252	43	𝑏𝑗	𝑏𝑗	PROPN
cana-843	252	44	,	,	PUNCT
cana-843	252	45	a	a	DET
cana-843	252	46	vertex	vertex	NOUN
cana-843	252	47	of	of	ADP
cana-843	252	48	𝐻𝑗	𝐻𝑗	NOUN
cana-843	252	49	,	,	PUNCT
cana-843	252	50	”	"	PUNCT
cana-843	252	51	is	be	AUX
cana-843	252	52	different	different	ADJ
cana-843	252	53	from	from	ADP
cana-843	252	54	𝑐𝑗.	𝑐𝑗.	NOUN
cana-843	252	55	following	follow	VERB
cana-843	252	56	that	that	PRON
cana-843	252	57	,	,	PUNCT
cana-843	252	58	𝑆1	𝑆1	NOUN
cana-843	252	59	=	=	SYM
cana-843	252	60	(	(	PUNCT
cana-843	252	61	𝑆	𝑆	PROPN
cana-843	252	62	−	−	PROPN
cana-843	252	63	{	{	PUNCT
cana-843	252	64	𝑐𝑗	𝑐𝑗	NOUN
cana-843	252	65	}	}	PUNCT
cana-843	252	66	)	)	PUNCT
cana-843	252	67	∪	∪	ADP
cana-843	252	68	{	{	PUNCT
cana-843	252	69	𝑏𝑗	𝑏𝑗	NOUN
cana-843	252	70	}	}	PUNCT
cana-843	252	71	is	be	AUX
cana-843	252	72	a	a	DET
cana-843	252	73	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	252	74	that	that	PRON
cana-843	252	75	correctly	correctly	ADV
cana-843	252	76	contains	contain	VERB
cana-843	252	77	e𝑇.	e𝑇.	ADJ
cana-843	252	78	such	such	ADJ
cana-843	252	79	that	that	SCONJ
cana-843	252	80	𝑇	𝑇	PROPN
cana-843	252	81	is	be	AUX
cana-843	252	82	not	not	PART
cana-843	252	83	a	a	DET
cana-843	252	84	forced	force	VERB
cana-843	252	85	subset	subset	NOUN
cana-843	252	86	of	of	ADP
cana-843	252	87	𝑆	𝑆	PROPN
cana-843	252	88	as	as	ADP
cana-843	252	89	a	a	DET
cana-843	252	90	result	result	NOUN
cana-843	252	91	.	.	PUNCT
cana-843	253	1	for	for	ADP
cana-843	253	2	every	every	DET
cana-843	253	3	minimum	minimum	ADJ
cana-843	253	4	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	253	5	of	of	ADP
cana-843	253	6	𝐺	𝐺	PROPN
cana-843	253	7	,	,	PUNCT
cana-843	253	8	this	this	PRON
cana-843	253	9	is	be	AUX
cana-843	253	10	true	true	ADJ
cana-843	253	11	.	.	PUNCT
cana-843	254	1	hence	hence	ADV
cana-843	254	2	,	,	PUNCT
cana-843	254	3	𝑓𝑐𝑟(𝐺	𝑓𝑐𝑟(𝐺	PROPN
cana-843	254	4	)	)	PUNCT
cana-843	255	1	=	=	SYM
cana-843	255	2	𝑎.	𝑎.	VERB
cana-843	255	3	we	we	PRON
cana-843	255	4	then	then	ADV
cana-843	255	5	demonstrate	demonstrate	VERB
cana-843	255	6	that	that	SCONJ
cana-843	255	7	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	255	8	)	)	PUNCT
cana-843	255	9	=	=	PUNCT
cana-843	255	10	𝑎.	𝑎.	NOUN
cana-843	255	11	let	let	VERB
cana-843	255	12	𝑍	𝑍	VERB
cana-843	255	13	=	=	SYM
cana-843	255	14	{	{	PUNCT
cana-843	255	15	𝑡1	𝑡1	NOUN
cana-843	255	16	,	,	PUNCT
cana-843	255	17	𝑡3	𝑡3	PROPN
cana-843	255	18	,	,	PUNCT
cana-843	255	19	𝑤1	𝑤1	VERB
cana-843	255	20	,	,	PUNCT
cana-843	255	21	𝑤3	𝑤3	PROPN
cana-843	255	22	}	}	PUNCT
cana-843	255	23	represent	represent	VERB
cana-843	255	24	all	all	PRON
cana-843	255	25	of	of	ADP
cana-843	255	26	𝐺	𝐺	PROPN
cana-843	255	27	's	's	PART
cana-843	255	28	extreme	extreme	ADJ
cana-843	255	29	vertices	vertex	NOUN
cana-843	255	30	.	.	PUNCT
cana-843	256	1	consider	consider	VERB
cana-843	256	2	𝑍	𝑍	PROPN
cana-843	256	3	is	be	AUX
cana-843	256	4	a	a	DET
cana-843	256	5	subset	subset	NOUN
cana-843	256	6	of	of	ADP
cana-843	256	7	each	each	DET
cana-843	256	8	geodetic	geodetic	ADJ
cana-843	256	9	set	set	NOUN
cana-843	256	10	in	in	ADP
cana-843	256	11	𝐺	𝐺	PROPN
cana-843	256	12	,	,	PUNCT
cana-843	256	13	according	accord	VERB
cana-843	256	14	to	to	ADP
cana-843	256	15	theorem	theorem	ADJ
cana-843	256	16	3.4	3.4	NUM
cana-843	256	17	.	.	PUNCT
cana-843	257	1	𝑍	𝑍	NOUN
cana-843	257	2	should	should	AUX
cana-843	257	3	be	be	AUX
cana-843	257	4	𝑍	𝑍	NOUN
cana-843	257	5	=	=	SYM
cana-843	257	6	𝑍1	𝑍1	NOUN
cana-843	257	7	∪	∪	X
cana-843	257	8	{	{	PUNCT
cana-843	257	9	𝑤	𝑤	NOUN
cana-843	257	10	}	}	PUNCT
cana-843	257	11	.	.	PUNCT
cana-843	258	1	therefore	therefore	ADV
cana-843	258	2	𝑍1	𝑍1	PROPN
cana-843	258	3	is	be	AUX
cana-843	258	4	clearly	clearly	ADV
cana-843	258	5	a	a	DET
cana-843	258	6	subset	subset	NOUN
cana-843	258	7	of	of	ADP
cana-843	258	8	each	each	PRON
cana-843	258	9	and	and	CCONJ
cana-843	258	10	every	every	DET
cana-843	258	11	geodetic	geodetic	ADJ
cana-843	258	12	set	set	NOUN
cana-843	258	13	in	in	ADP
cana-843	258	14	𝐺.	𝐺.	NOUN
cana-843	258	15	assume	assume	VERB
cana-843	258	16	𝑄𝑗	𝑄𝑗	PROPN
cana-843	258	17	:	:	PUNCT
cana-843	258	18	{	{	PUNCT
cana-843	258	19	𝑢𝑗	𝑢𝑗	NOUN
cana-843	258	20	,	,	PUNCT
cana-843	258	21	𝑣𝑗	𝑣𝑗	ADP
cana-843	258	22	}	}	PUNCT
cana-843	258	23	(	(	PUNCT
cana-843	258	24	1	1	NUM
cana-843	258	25	≤	≤	NUM
cana-843	258	26	𝑗	𝑗	PRON
cana-843	258	27	≤	≤	NUM
cana-843	259	1	𝑎	𝑎	PRON
cana-843	259	2	−	−	NOUN
cana-843	259	3	𝑏	𝑏	NOUN
cana-843	259	4	)	)	PUNCT
cana-843	259	5	.	.	PUNCT
cana-843	260	1	every	every	DET
cana-843	260	2	geodetic	geodetic	ADJ
cana-843	260	3	set	set	NOUN
cana-843	260	4	of	of	ADP
cana-843	260	5	𝐺	𝐺	PROPN
cana-843	260	6	has	have	VERB
cana-843	260	7	at	at	ADV
cana-843	260	8	least	least	ADV
cana-843	260	9	one	one	NUM
cana-843	260	10	vertex	vertex	NOUN
cana-843	260	11	from	from	ADP
cana-843	260	12	each	each	PRON
cana-843	260	13	of	of	ADP
cana-843	260	14	the	the	DET
cana-843	260	15	𝐻𝑗	𝐻𝑗	PROPN
cana-843	260	16	(	(	PUNCT
cana-843	260	17	1	1	NUM
cana-843	260	18	≤	≤	NUM
cana-843	260	19	𝑗	𝑗	PRON
cana-843	260	20	≤	≤	NUM
cana-843	260	21	𝑎	𝑎	NOUN
cana-843	260	22	)	)	PUNCT
cana-843	260	23	and	and	CCONJ
cana-843	260	24	each	each	PRON
cana-843	260	25	of	of	ADP
cana-843	260	26	the	the	DET
cana-843	260	27	𝑄𝑗	𝑄𝑗	PROPN
cana-843	260	28	,	,	PUNCT
cana-843	260	29	as	as	SCONJ
cana-843	260	30	can	can	AUX
cana-843	260	31	be	be	AUX
cana-843	260	32	clearly	clearly	ADV
cana-843	260	33	recognised	recognise	VERB
cana-843	260	34	;	;	PUNCT
cana-843	260	35	so	so	ADV
cana-843	260	36	,	,	PUNCT
cana-843	260	37	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	260	38	)	)	PUNCT
cana-843	260	39	≥	≥	NOUN
cana-843	260	40	4	4	NUM
cana-843	261	1	+	+	CCONJ
cana-843	261	2	𝑎	𝑎	X
cana-843	261	3	−	−	NOUN
cana-843	261	4	𝑏	𝑏	NOUN
cana-843	261	5	+	+	CCONJ
cana-843	261	6	𝑏	𝑏	NOUN
cana-843	261	7	=	=	SYM
cana-843	261	8	4	4	NUM
cana-843	261	9	+	+	NUM
cana-843	261	10	𝑎.	𝑎.	NOUN
cana-843	261	11	assume	assume	VERB
cana-843	261	12	𝑊	𝑊	PROPN
cana-843	261	13	=	=	NOUN
cana-843	261	14	𝑍1	𝑍1	PROPN
cana-843	261	15	∪	∪	X
cana-843	261	16	{	{	PUNCT
cana-843	261	17	𝑟1	𝑟1	NOUN
cana-843	261	18	,	,	PUNCT
cana-843	261	19	𝑟2	𝑟2	NOUN
cana-843	261	20	,	,	PUNCT
cana-843	261	21	…	…	PUNCT
cana-843	261	22	,	,	PUNCT
cana-843	261	23	𝑟𝑏	𝑟𝑏	INTJ
cana-843	261	24	,	,	PUNCT
cana-843	261	25	𝑢1	𝑢1	PROPN
cana-843	261	26	,	,	PUNCT
cana-843	261	27	𝑢2	𝑢2	PROPN
cana-843	261	28	,	,	PUNCT
cana-843	261	29	…	…	PUNCT
cana-843	261	30	,	,	PUNCT
cana-843	261	31	𝑢𝑎−𝑏	𝑢𝑎−𝑏	PROPN
cana-843	261	32	}	}	PUNCT
cana-843	261	33	.	.	PUNCT
cana-843	262	1	consequently	consequently	ADV
cana-843	262	2	,	,	PUNCT
cana-843	262	3	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	262	4	)	)	PUNCT
cana-843	262	5	=	=	PUNCT
cana-843	263	1	𝑎	𝑎	X
cana-843	263	2	+	+	NUM
cana-843	263	3	4	4	NUM
cana-843	263	4	since	since	SCONJ
cana-843	263	5	𝐼[𝑊	𝐼[𝑊	ADJ
cana-843	263	6	]	]	X
cana-843	263	7	=	=	SYM
cana-843	263	8	𝑉(𝐺	𝑉(𝐺	NOUN
cana-843	263	9	)	)	PUNCT
cana-843	263	10	and	and	CCONJ
cana-843	263	11	𝑊	𝑊	PROPN
cana-843	263	12	is	be	AUX
cana-843	263	13	a	a	DET
cana-843	263	14	geodetic	geodetic	ADJ
cana-843	263	15	set	set	NOUN
cana-843	263	16	of	of	ADP
cana-843	263	17	𝐺.	𝐺.	NOUN
cana-843	263	18	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	263	19	)	)	PUNCT
cana-843	264	1	≤	≤	NUM
cana-843	264	2	𝑔(𝐺	𝑔(𝐺	NOUN
cana-843	264	3	)	)	PUNCT
cana-843	265	1	−	−	NOUN
cana-843	265	2	|𝑍|	|𝑍|	NOUN
cana-843	266	1	=	=	PUNCT
cana-843	267	1	𝑎	𝑎	X
cana-843	267	2	+	+	NUM
cana-843	267	3	4	4	NUM
cana-843	267	4	−	−	NOUN
cana-843	267	5	4	4	NUM
cana-843	267	6	=	=	SYM
cana-843	267	7	𝑎	𝑎	NOUN
cana-843	267	8	according	accord	VERB
cana-843	267	9	to	to	ADP
cana-843	267	10	theorem	theorem	NOUN
cana-843	267	11	1.2	1.2	NUM
cana-843	267	12	.	.	PUNCT
cana-843	268	1	it	it	PRON
cana-843	268	2	is	be	AUX
cana-843	268	3	evident	evident	ADJ
cana-843	268	4	that	that	SCONJ
cana-843	268	5	every	every	DET
cana-843	268	6	𝑔set	𝑔set	NOUN
cana-843	268	7	of	of	ADP
cana-843	268	8	𝐺	𝐺	PROPN
cana-843	268	9	is	be	AUX
cana-843	268	10	of	of	ADP
cana-843	268	11	the	the	DET
cana-843	268	12	form	form	NOUN
cana-843	268	13	𝑊1	𝑊1	PROPN
cana-843	269	1	=	=	SYM
cana-843	269	2	𝑍	𝑍	PROPN
cana-843	269	3	∪	∪	ADJ
cana-843	269	4	{	{	PUNCT
cana-843	269	5	e𝑐1	e𝑐1	NOUN
cana-843	269	6	,	,	PUNCT
cana-843	269	7	e𝑐2	e𝑐2	PROPN
cana-843	269	8	,	,	PUNCT
cana-843	269	9	…	…	PUNCT
cana-843	269	10	,	,	PUNCT
cana-843	269	11	e𝑐𝑏	e𝑐𝑏	PROPN
cana-843	269	12	}	}	PUNCT
cana-843	269	13	∪	∪	ADJ
cana-843	269	14	{	{	PUNCT
cana-843	269	15	𝑑1	𝑑1	NOUN
cana-843	269	16	,	,	PUNCT
cana-843	269	17	𝑑2	𝑑2	NOUN
cana-843	269	18	,	,	PUNCT
cana-843	269	19	…	…	PUNCT
cana-843	269	20	,	,	PUNCT
cana-843	269	21	𝑑𝑎−𝑏	𝑑𝑎−𝑏	PROPN
cana-843	269	22	}	}	PUNCT
cana-843	269	23	since	since	SCONJ
cana-843	269	24	𝑍	𝑍	PROPN
cana-843	269	25	is	be	AUX
cana-843	269	26	a	a	DET
cana-843	269	27	subset	subset	NOUN
cana-843	269	28	of	of	ADP
cana-843	269	29	every	every	DET
cana-843	269	30	𝑔-set	𝑔-set	NOUN
cana-843	269	31	of	of	ADP
cana-843	269	32	e𝐺.	e𝐺.	NOUN
cana-843	269	33	in	in	ADP
cana-843	269	34	this	this	DET
cana-843	269	35	case	case	NOUN
cana-843	269	36	,	,	PUNCT
cana-843	270	1	𝑑𝑗	𝑑𝑗	PROPN
cana-843	270	2	∈	∈	PROPN
cana-843	270	3	𝑄𝑗	𝑄𝑗	PROPN
cana-843	270	4	(	(	PUNCT
cana-843	270	5	1	1	NUM
cana-843	270	6	≤	≤	NUM
cana-843	270	7	𝑗	𝑗	PRON
cana-843	270	8	≤	≤	NUM
cana-843	270	9	𝑎	𝑎	PRON
cana-843	270	10	−	−	NOUN
cana-843	270	11	𝑏	𝑏	NOUN
cana-843	270	12	)	)	PUNCT
cana-843	270	13	and	and	CCONJ
cana-843	270	14	𝑐𝑖	𝑐𝑖	NOUN
cana-843	270	15	∈	∈	PROPN
cana-843	271	1	𝐻𝑖	𝐻𝑖	PROPN
cana-843	271	2	(	(	PUNCT
cana-843	271	3	1	1	NUM
cana-843	271	4	≤	≤	NUM
cana-843	271	5	𝑖	𝑖	SYM
cana-843	271	6	≤	≤	NOUN
cana-843	271	7	𝑏	𝑏	NOUN
cana-843	271	8	)	)	PUNCT
cana-843	271	9	.	.	PUNCT
cana-843	272	1	given	give	VERB
cana-843	272	2	|𝑇|	|𝑇|	NOUN
cana-843	272	3	<	<	X
cana-843	272	4	𝑏	𝑏	NOUN
cana-843	272	5	,	,	PUNCT
cana-843	272	6	let	let	VERB
cana-843	272	7	𝑇	𝑇	PROPN
cana-843	272	8	be	be	AUX
cana-843	272	9	any	any	DET
cana-843	272	10	proper	proper	ADJ
cana-843	272	11	subset	subset	NOUN
cana-843	272	12	of	of	ADP
cana-843	272	13	𝑊1	𝑊1	PROPN
cana-843	272	14	.	.	PUNCT
cana-843	273	1	after	after	ADP
cana-843	273	2	that	that	PRON
cana-843	273	3	,	,	PUNCT
cana-843	273	4	𝑐𝑗	𝑐𝑗	INTJ
cana-843	273	5	,	,	PUNCT
cana-843	273	6	𝑑𝑗	𝑑𝑗	PROPN
cana-843	273	7	∉	∉	PROPN
cana-843	273	8	𝑇	𝑇	PROPN
cana-843	273	9	since	since	SCONJ
cana-843	273	10	there	there	PRON
cana-843	273	11	are	be	VERB
cana-843	273	12	vertices	vertex	NOUN
cana-843	273	13	𝑐𝑖	𝑐𝑖	ADP
cana-843	273	14	∈	∈	PROPN
cana-843	273	15	𝐻𝑖	𝐻𝑖	PROPN
cana-843	273	16	and	and	CCONJ
cana-843	273	17	𝑑𝑗	𝑑𝑗	ADP
cana-843	273	18	∈	∈	PROPN
cana-843	274	1	𝑄𝑗.	𝑄𝑗.	PROPN
cana-843	274	2	assume	assume	VERB
cana-843	274	3	𝑄𝑗	𝑄𝑗	PROPN
cana-843	274	4	is	be	AUX
cana-843	274	5	a	a	DET
cana-843	274	6	vertex	vertex	NOUN
cana-843	274	7	of	of	ADP
cana-843	274	8	𝐻𝑖	𝐻𝑖	PROPN
cana-843	274	9	that	that	PRON
cana-843	274	10	is	be	AUX
cana-843	274	11	different	different	ADJ
cana-843	274	12	from	from	ADP
cana-843	274	13	𝑑𝑗	𝑑𝑗	PRON
cana-843	274	14	and	and	CCONJ
cana-843	274	15	𝑐𝑖.	𝑐𝑖.	NOUN
cana-843	274	16	𝑊2	𝑊2	PROPN
cana-843	274	17	=	=	SYM
cana-843	274	18	(	(	PUNCT
cana-843	274	19	𝑊1	𝑊1	PROPN
cana-843	274	20	−	−	PROPN
cana-843	275	1	{	{	PUNCT
cana-843	275	2	𝑐𝑗	𝑐𝑗	PROPN
cana-843	275	3	,	,	PUNCT
cana-843	275	4	𝑑𝑗	𝑑𝑗	NOUN
cana-843	275	5	}	}	PUNCT
cana-843	275	6	)	)	PUNCT
cana-843	275	7	∪	∪	ADP
cana-843	275	8	{	{	PUNCT
cana-843	275	9	𝑒𝑗	𝑒𝑗	NOUN
cana-843	275	10	,	,	PUNCT
cana-843	275	11	𝑓𝑗	𝑓𝑗	ADJ
cana-843	275	12	}	}	PUNCT
cana-843	275	13	is	be	AUX
cana-843	275	14	a	a	DET
cana-843	275	15	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	275	16	that	that	PRON
cana-843	275	17	correctly	correctly	ADV
cana-843	275	18	contains	contain	VERB
cana-843	275	19	𝑇	𝑇	PROPN
cana-843	275	20	in	in	ADP
cana-843	275	21	this	this	DET
cana-843	275	22	case	case	NOUN
cana-843	275	23	.	.	PUNCT
cana-843	276	1	such	such	ADJ
cana-843	276	2	that	that	SCONJ
cana-843	276	3	𝑇	𝑇	PROPN
cana-843	276	4	is	be	AUX
cana-843	276	5	not	not	PART
cana-843	276	6	a	a	DET
cana-843	276	7	forced	force	VERB
cana-843	276	8	subset	subset	NOUN
cana-843	276	9	of	of	ADP
cana-843	276	10	𝑆	𝑆	PROPN
cana-843	276	11	as	as	ADP
cana-843	276	12	a	a	DET
cana-843	276	13	result	result	NOUN
cana-843	276	14	.	.	PUNCT
cana-843	277	1	for	for	ADP
cana-843	277	2	every	every	DET
cana-843	277	3	minimum	minimum	ADJ
cana-843	277	4	𝑐𝑟-set	𝑐𝑟-set	NOUN
cana-843	277	5	of	of	ADP
cana-843	277	6	𝐺	𝐺	PROPN
cana-843	277	7	,	,	PUNCT
cana-843	277	8	this	this	PRON
cana-843	277	9	is	be	AUX
cana-843	277	10	true	true	ADJ
cana-843	277	11	.	.	PUNCT
cana-843	278	1	therefore	therefore	ADV
cana-843	278	2	,	,	PUNCT
cana-843	278	3	𝑓𝑔(𝐺	𝑓𝑔(𝐺	NOUN
cana-843	278	4	)	)	PUNCT
cana-843	278	5	=	=	PUNCT
cana-843	278	6	𝑎.	𝑎.	NOUN
cana-843	278	7	figure	figure	VERB
cana-843	278	8	3.6	3.6	NUM
cana-843	278	9	communications	communication	NOUN
cana-843	278	10	on	on	ADP
cana-843	278	11	applied	apply	VERB
cana-843	278	12	nonlinear	nonlinear	ADJ
cana-843	278	13	analysis	analysis	NOUN
cana-843	278	14	issn	issn	NOUN
cana-843	278	15	:	:	PUNCT
cana-843	278	16	1074	1074	NUM
cana-843	278	17	-	-	PUNCT
cana-843	278	18	133x	133x	NUM
cana-843	278	19	vol	vol	NOUN
cana-843	278	20	31	31	NUM
cana-843	278	21	no	no	NOUN
cana-843	278	22	.	.	PUNCT
cana-843	279	1	4s	4s	NUM
cana-843	279	2	(	(	PUNCT
cana-843	279	3	2024	2024	NUM
cana-843	279	4	)	)	PUNCT
cana-843	279	5	228	228	NUM
cana-843	279	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-843	279	7	references	reference	NOUN
cana-843	279	8	[	[	X
cana-843	279	9	1	1	NUM
cana-843	279	10	]	]	X
cana-843	279	11	f.	f.	PROPN
cana-843	279	12	buckley	buckley	PROPN
cana-843	279	13	and	and	CCONJ
cana-843	279	14	f.	f.	PROPN
cana-843	279	15	harary	harary	PROPN
cana-843	279	16	,	,	PUNCT
cana-843	279	17	distance	distance	NOUN
cana-843	279	18	in	in	ADP
cana-843	279	19	graphs	graph	NOUN
cana-843	279	20	,	,	PUNCT
cana-843	279	21	addision	addision	NOUN
cana-843	279	22	-	-	PUNCT
cana-843	279	23	weseely	weseely	ADV
cana-843	279	24	,	,	PUNCT
cana-843	279	25	reading	read	VERB
cana-843	279	26	ma	ma	PROPN
cana-843	279	27	,	,	PUNCT
cana-843	279	28	(	(	PUNCT
cana-843	279	29	1990	1990	NUM
cana-843	279	30	)	)	PUNCT
cana-843	279	31	.	.	PUNCT
cana-843	280	1	[	[	X
cana-843	280	2	2	2	X
cana-843	280	3	]	]	X
cana-843	280	4	g.	g.	PROPN
cana-843	280	5	chartrand	chartrand	PROPN
cana-843	280	6	and	and	CCONJ
cana-843	280	7	p.	p.	PROPN
cana-843	280	8	zhang	zhang	PROPN
cana-843	280	9	,	,	PUNCT
cana-843	280	10	the	the	DET
cana-843	280	11	forcing	force	VERB
cana-843	280	12	geodetic	geodetic	ADJ
cana-843	280	13	number	number	NOUN
cana-843	280	14	of	of	ADP
cana-843	280	15	a	a	DET
cana-843	280	16	graph	graph	NOUN
cana-843	280	17	,	,	PUNCT
cana-843	280	18	discuss	discuss	NOUN
cana-843	280	19	.	.	PUNCT
cana-843	281	1	graph	graph	NOUN
cana-843	281	2	theory	theory	NOUN
cana-843	281	3	,	,	PUNCT
cana-843	281	4	19	19	NUM
cana-843	281	5	,	,	PUNCT
cana-843	281	6	(	(	PUNCT
cana-843	281	7	1999	1999	NUM
cana-843	281	8	)	)	PUNCT
cana-843	281	9	,	,	PUNCT
cana-843	281	10	45	45	NUM
cana-843	281	11	-	-	SYM
cana-843	281	12	58	58	NUM
cana-843	281	13	.	.	PUNCT
cana-843	282	1	[	[	X
cana-843	282	2	3	3	X
cana-843	282	3	]	]	X
cana-843	282	4	g.	g.	PROPN
cana-843	282	5	chartrand	chartrand	PROPN
cana-843	282	6	,	,	PUNCT
cana-843	282	7	f.	f.	PROPN
cana-843	282	8	harary	harary	PROPN
cana-843	282	9	and	and	CCONJ
cana-843	282	10	p.	p.	PROPN
cana-843	282	11	zhang	zhang	PROPN
cana-843	282	12	,	,	PUNCT
cana-843	282	13	on	on	ADP
cana-843	282	14	the	the	DET
cana-843	282	15	geodetic	geodetic	ADJ
cana-843	282	16	number	number	NOUN
cana-843	282	17	of	of	ADP
cana-843	282	18	a	a	DET
cana-843	282	19	graph	graph	NOUN
cana-843	282	20	,	,	PUNCT
cana-843	282	21	networks	network	NOUN
cana-843	282	22	,	,	PUNCT
cana-843	282	23	39(1	39(1	NUM
cana-843	282	24	)	)	PUNCT
cana-843	282	25	,	,	PUNCT
cana-843	282	26	(	(	PUNCT
cana-843	282	27	2002	2002	NUM
cana-843	282	28	)	)	PUNCT
cana-843	282	29	,	,	PUNCT
cana-843	282	30	1	1	NUM
cana-843	282	31	6	6	NUM
cana-843	282	32	.	.	PUNCT
cana-843	283	1	[	[	X
cana-843	283	2	4	4	X
cana-843	283	3	]	]	X
cana-843	283	4	g.	g.	PROPN
cana-843	283	5	chartrand	chartrand	PROPN
cana-843	283	6	,	,	PUNCT
cana-843	283	7	e.	e.	PROPN
cana-843	283	8	m.	m.	PROPN
cana-843	283	9	palmer	palmer	PROPN
cana-843	283	10	and	and	CCONJ
cana-843	283	11	p.	p.	PROPN
cana-843	283	12	zhang	zhang	PROPN
cana-843	283	13	,	,	PUNCT
cana-843	283	14	the	the	DET
cana-843	283	15	geodetic	geodetic	ADJ
cana-843	283	16	number	number	NOUN
cana-843	283	17	of	of	ADP
cana-843	283	18	a	a	DET
cana-843	283	19	graph	graph	NOUN
cana-843	283	20	,	,	PUNCT
cana-843	283	21	a	a	DET
cana-843	283	22	survey	survey	NOUN
cana-843	283	23	,	,	PUNCT
cana-843	283	24	congressus	congressus	PROPN
cana-843	283	25	numerantium	numerantium	PROPN
cana-843	283	26	,	,	PUNCT
cana-843	283	27	156	156	NUM
cana-843	283	28	,	,	PUNCT
cana-843	283	29	(	(	PUNCT
cana-843	283	30	2002	2002	NUM
cana-843	283	31	)	)	PUNCT
cana-843	283	32	,	,	PUNCT
cana-843	283	33	37	37	NUM
cana-843	283	34	58	58	NUM
cana-843	283	35	.	.	PUNCT
cana-843	284	1	[	[	X
cana-843	284	2	5	5	X
cana-843	284	3	]	]	PUNCT
cana-843	284	4	g.	g.	PROPN
cana-843	284	5	chartrand	chartrand	PROPN
cana-843	284	6	,	,	PUNCT
cana-843	284	7	l.	l.	PROPN
cana-843	284	8	johns	johns	PROPN
cana-843	284	9	and	and	CCONJ
cana-843	284	10	p.	p.	PROPN
cana-843	284	11	zang	zang	PROPN
cana-843	284	12	,	,	PUNCT
cana-843	284	13	detour	detour	PRON
cana-843	284	14	number	number	NOUN
cana-843	284	15	of	of	ADP
cana-843	284	16	graph	graph	NOUN
cana-843	284	17	,	,	PUNCT
cana-843	284	18	utilitas	utilitas	ADJ
cana-843	284	19	mathematics	mathematic	NOUN
cana-843	284	20	,	,	PUNCT
cana-843	284	21	64	64	NUM
cana-843	284	22	(	(	PUNCT
cana-843	284	23	2003	2003	NUM
cana-843	284	24	)	)	PUNCT
cana-843	284	25	,	,	PUNCT
cana-843	284	26	97	97	NUM
cana-843	284	27	-	-	SYM
cana-843	284	28	113	113	NUM
cana-843	284	29	.	.	PUNCT
cana-843	285	1	[	[	X
cana-843	285	2	6	6	NUM
cana-843	285	3	]	]	PUNCT
cana-843	285	4	g.	g.	PROPN
cana-843	285	5	chartrand	chartrand	PROPN
cana-843	285	6	,	,	PUNCT
cana-843	285	7	h.	h.	PROPN
cana-843	285	8	escuadro	escuadro	PROPN
cana-843	285	9	and	and	CCONJ
cana-843	285	10	p.	p.	PROPN
cana-843	285	11	zhang	zhang	PROPN
cana-843	285	12	,	,	PUNCT
cana-843	285	13	distance	distance	NOUN
cana-843	285	14	in	in	ADP
cana-843	285	15	graphs	graph	NOUN
cana-843	285	16	,	,	PUNCT
cana-843	285	17	taking	take	VERB
cana-843	285	18	the	the	DET
cana-843	285	19	long	long	ADJ
cana-843	285	20	view	view	NOUN
cana-843	285	21	,	,	PUNCT
cana-843	285	22	akce	akce	PROPN
cana-843	285	23	j.	j.	PROPN
cana-843	285	24	graphs	graphs	PROPN
cana-843	285	25	and	and	CCONJ
cana-843	285	26	combin	combin	NOUN
cana-843	285	27	.	.	PROPN
cana-843	285	28	,	,	PUNCT
cana-843	285	29	1(1	1(1	NUM
cana-843	285	30	)	)	PUNCT
cana-843	285	31	(	(	PUNCT
cana-843	285	32	2004	2004	NUM
cana-843	285	33	)	)	PUNCT
cana-843	285	34	,	,	PUNCT
cana-843	285	35	1	1	NUM
cana-843	285	36	-	-	SYM
cana-843	285	37	13	13	NUM
cana-843	285	38	.	.	PUNCT
cana-843	286	1	[	[	X
cana-843	286	2	7	7	X
cana-843	286	3	]	]	X
cana-843	286	4	g.	g.	PROPN
cana-843	286	5	chartrand	chartrand	PROPN
cana-843	286	6	,	,	PUNCT
cana-843	286	7	h.	h.	PROPN
cana-843	286	8	escuadro	escuadro	PROPN
cana-843	286	9	and	and	CCONJ
cana-843	286	10	b.	b.	PROPN
cana-843	286	11	zang	zang	PROPN
cana-843	286	12	,	,	PUNCT
cana-843	286	13	detour	detour	NOUN
cana-843	286	14	distance	distance	NOUN
cana-843	286	15	in	in	ADP
cana-843	286	16	graph	graph	NOUN
cana-843	286	17	,	,	PUNCT
cana-843	286	18	j.	j.	PROPN
cana-843	286	19	combin	combin	PROPN
cana-843	286	20	,	,	PUNCT
cana-843	286	21	mathcombin	mathcombin	NOUN
cana-843	286	22	,	,	PUNCT
cana-843	286	23	compul	compul	NOUN
cana-843	286	24	53	53	NUM
cana-843	286	25	(	(	PUNCT
cana-843	286	26	2005	2005	NUM
cana-843	286	27	)	)	PUNCT
cana-843	286	28	75	75	NUM
cana-843	286	29	-	-	SYM
cana-843	286	30	94	94	NUM
cana-843	286	31	.	.	PUNCT
cana-843	287	1	[	[	X
cana-843	287	2	8	8	NUM
cana-843	287	3	]	]	PUNCT
cana-843	287	4	a.	a.	NOUN
cana-843	287	5	hansberg	hansberg	PROPN
cana-843	287	6	,	,	PUNCT
cana-843	287	7	l.	l.	PROPN
cana-843	287	8	volkmann	volkmann	PROPN
cana-843	287	9	,	,	PUNCT
cana-843	287	10	on	on	ADP
cana-843	287	11	the	the	DET
cana-843	287	12	geodetic	geodetic	ADJ
cana-843	287	13	and	and	CCONJ
cana-843	287	14	geodetic	geodetic	ADJ
cana-843	287	15	domination	domination	NOUN
cana-843	287	16	numbers	number	NOUN
cana-843	287	17	of	of	ADP
cana-843	287	18	a	a	DET
cana-843	287	19	graph	graph	NOUN
cana-843	287	20	,	,	PUNCT
cana-843	287	21	discrete	discrete	ADJ
cana-843	287	22	mathematics	mathematic	NOUN
cana-843	287	23	,	,	PUNCT
cana-843	287	24	310	310	NUM
cana-843	287	25	(	(	PUNCT
cana-843	287	26	15	15	NUM
cana-843	287	27	-	-	SYM
cana-843	287	28	16	16	NUM
cana-843	287	29	)	)	PUNCT
cana-843	287	30	,	,	PUNCT
cana-843	287	31	(	(	PUNCT
cana-843	287	32	2010	2010	NUM
cana-843	287	33	)	)	PUNCT
cana-843	287	34	,	,	PUNCT
cana-843	287	35	2140	2140	NUM
cana-843	287	36	-	-	SYM
cana-843	287	37	2146	2146	NUM
cana-843	287	38	.	.	PUNCT
cana-843	288	1	[	[	X
cana-843	288	2	9	9	NUM
cana-843	288	3	]	]	PUNCT
cana-843	288	4	p.	p.	NOUN
cana-843	288	5	lakshmi	lakshmi	PROPN
cana-843	288	6	narayana	narayana	PROPN
cana-843	288	7	varma	varma	PROPN
cana-843	288	8	and	and	CCONJ
cana-843	288	9	j.	j.	PROPN
cana-843	288	10	veeranjaneyulu	veeranjaneyulu	PROPN
cana-843	288	11	,	,	PUNCT
cana-843	288	12	study	study	NOUN
cana-843	288	13	of	of	ADP
cana-843	288	14	circular	circular	ADJ
cana-843	288	15	distance	distance	NOUN
cana-843	288	16	in	in	ADP
cana-843	288	17	graphs	graph	NOUN
cana-843	288	18	,	,	PUNCT
cana-843	288	19	turkish	turkish	ADJ
cana-843	288	20	journal	journal	NOUN
cana-843	288	21	of	of	ADP
cana-843	288	22	computer	computer	NOUN
cana-843	288	23	and	and	CCONJ
cana-843	288	24	mathematics	mathematic	NOUN
cana-843	288	25	education	education	NOUN
cana-843	288	26	12(2	12(2	NUM
cana-843	288	27	)	)	PUNCT
cana-843	288	28	,	,	PUNCT
cana-843	288	29	(	(	PUNCT
cana-843	288	30	2021),2437	2021),2437	NUM
cana-843	288	31	-	-	SYM
cana-843	288	32	2444	2444	NUM
cana-843	288	33	.	.	PUNCT
cana-843	289	1	[	[	X
cana-843	289	2	10	10	NUM
cana-843	289	3	]	]	X
cana-843	289	4	s.	s.	PROPN
cana-843	289	5	sheeja	sheeja	PROPN
cana-843	289	6	and	and	CCONJ
cana-843	289	7	k.	k.	PROPN
cana-843	289	8	rajendran	rajendran	PROPN
cana-843	289	9	,	,	PUNCT
cana-843	289	10	the	the	DET
cana-843	289	11	circular	circular	ADJ
cana-843	289	12	number	number	NOUN
cana-843	289	13	of	of	ADP
cana-843	289	14	a	a	DET
cana-843	289	15	graph	graph	NOUN
cana-843	289	16	(	(	PUNCT
cana-843	289	17	communicated	communicate	VERB
cana-843	289	18	)	)	PUNCT
cana-843	289	19	.	.	PUNCT